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IP . a i unoiis < \f( uni'tirtm e the calculation ol (hani>,e- 
\\he<li foi jlir tuttin*; t>( diffeient pitches ol thiead on 
a lathe, hmu'vet simple -.itch ,i udt ulaliojp may he, is 
u>jnp.itativel\ but little Known, hc-int;, lor the tn.ijoii'ty ol 
Ihosv most ( loscly intctrstal in ttu* suhjc'ct, shioiuletl HI 
in>stt'i v 

Many \\hosr thcoK'lual knowl'<ljt' is quite 1 suliuicnl 
to jMKihlt tht'tn lo face the problem, have had so little 
jnaetual expenenre in setew-uitttng tliey .u'e uiiiiblc 
to to <kej)ly into Ihe m.ittet, and pie.sent, in a eleui (in<l 
siiiiplr inatHH'i, the tlifietenl variations whiuh in.iy possibly 

The \\n ,tlei nutubei ot ineehanics, even the ytuui;>ei ones, 
pu.-.tss (no sh;hl a the(*retltil knowledge to permit ol then 
btuhlui", up a -.ystein by themselves. 

thru aie, ol ionise, iiHHhaiuch who are quit< j capable 
o{ uoikui^', tnit tlic iH'fu.ssiuy rukulation, but so many of 
them I speak ftom petsonal experience regard their 
knowl<*d!' ( e an aunt* or less of n secret, and uy, ut any rale 
to theniM'lvi"., "Why should 1 impart to others what ha*', 
taken tut 4 M ituu'h trouble and cost me so much money 
lo Irani j 1 *' 

The pin pose ol the present treatise is to unable any 
one, who Is ptepared to take tine trouble to study it carefully 
U Irani how to *ah ulate change- wheels piopeily, 

vi The Calculation oj Change- Wheel*. 

I have deemed it expedient, foi the sake of those of 
my readers who have but a supeificial knowledge oi tlu- 
lathe, to give a shoit desenption of this tool, in so fat as it 
is connected with sciew-cuttmg, to winch 1 have atUeil a 
de&ciiption of the various types of thiuad to be met with, 
with the necessaiy tables appended, as also a nunihet ui 
piactical hints, with icfcience to anew-cutting, together with 
the operations connected theiewith. 

I have puiposcly refiained from including, 1 ; a munhei ( 
tables giving the change-wheels requited for the v.uiuun 
pitches of thicads on different lathen, in place of whuh a 
large numbci of piactical examples aie given which cuvei 
eveiy possible variation likely to be met with in practical 
work. Experience has taught me that the inclusitm o! such 
tables only leads to purely mechanical work demanding 
no effort of the mind, whereas, in each partieuhu c.ue, tlu 
consideration should be given to the special work in ham!, 
so that in cases of exceptional difficulty, whore one is obliged 
to set to woik without the assistance of such tables, the 
manner of calculation may not be unfamiliar, 

It is my earnest wish that the present work may prow 
useful not only to students, but also to those engaged in 
practical work, 







THF LA mi , , , .. , y 


{/*) Systems ,, ,, , ,, .. no 

(/*) Whflt C "ha nt>e Wheels aic to he found on a Lathe . 15 

{/) Thf Cutting ol Metrie Thieuds on a Lathe with Motile 

Leadheiew , ,. , . . , 16 

(/} The Cutting of Mni'ltJih Threw Is on a Lathe with 

iCfij'hhh l,e,ld,seiew .. . . ,, ,, 3O 
if) To C*ttt Kiifjltsh Thread?, on u I'ilh< i with Metric 

(/) Tin C'utiini 1 , ol Metric Thiends i>n a Lathe with 

tyiglihh Lesidwttnv ,, ,, , 37 

(,C) Thr Wheel with is? Teeth ,, ., , . ,, 30 

{//} Method foi Caieulnting Appioximalo {''tnefions ., 32 

{/) The I 'tool of the Sum , ,, , ,, ,. 51 

{&) i\ nm up the Wheels , .. , 53 

{/) Thrfc'uftmj, with Double Compound Train 5/1 

(i) TJu? Cutting oi* Left-hand Threads ,, ,, ,, 55 

vin The Calculation of Change- Wheels 



(a) Foims of Thieads 56 

(b} Types of Thieads 56 

(c) Screw-Cutting Tools 65 

(J) Cutting the Thiead 6<; 
((?) The Cutting of Double and Multiple Thi ended Sacws 73 

(/) The Cutting of very Coaise Thiead 75 

(g) The Hendey-Noilon System 77 







TliRKAHS, both internal and external, can be obtained in two 
different ways, the simplest of which is to cut the thread by 
means of taps, dies and chaseis. In the smaller sizes, the 
majority of internal threads aie tapped, whilst external threads 
are cut with dies, but in the larger sizes too much material has 
to be removed. Tapping, however, is far moi e general than the 
use of dies, as in most cases, external tin cads can be obtained 
in anothei way, viz. : on the lathe, whilst internal threads 
can only be obtained on the lathe at considerable expense, 
M 01 cover, internal threads are to be found in a number ol 
different places on the largei machine parts, and so it would 
be well-nigh impossible to put these pieces on the lathe foi 
the puipose of cutting the threads. On the othei hand, a 
bolt 01 screw-spindle, as a rule, can be set on the lathe, and 
thieads may be cut by means of a common tool. It is just 
for this reason that, whilst a large number of i in. external 
thieads are cut on the lathe, I in. threads in holes aic, with 
but Few exceptions, cut exclusively by tapping. The practice, 
however, of cutting internal threads of more than 2 in. 
diameter on the lathe, whenever the work-piece allows it, is 
becoming more and more general. 

The object of the present work is to give a detailed 
description of the way in which it is possible to cut the various 


2 The Calculation of Change-Wheeh 

thieads on the lathe, and thus to answer, as fully as possible, 
the question " How are the change-wheels to be calculated 
foi screw-cutting on the lathe ? " 

In order that this woik may also be of service to those who 
are not fully conversant with the lathe, the following points 
will be treated successively, viz the gcneial constiuction of 
the lathe, moie especially of those paits of the Lithe used in 
screw-cutting , the theory of the calculation of change-wheels 
and screw-cutting m practice 

FIG. i 

The lathe, as originally constructed, was not intended foi 
sciew-cutting Fig i shows a lathe as it was fiist constructed. 
On this lathe a rotary movement was imparted by means of 
a driving belt to the headstock and workpicce only, all other 
movements being executed by the operator himself. 

Within a compaiatively short time, however, more wa.s 
demanded of this machine, larger pieces were required to be 
machined than was possible wuh direct belt drive, and the 

for Screw-cuffing on Lathes, 


double back t*vai was mtioduced , it was deshed to move the 
tool on the mateua.1 automatically, and to obtain this, the 
iest was mounted on a carriage and moved by means of a 
leadscrew which motion was impaitod by means of eithet 
a belt or a iiam of ^eais fiom the hcadstock. The intto- 
duction of a train of gems on the apt on made it possible 
not only to move the cairiage ovej the whole length of the 
bed foi sliding, bul also to move tlie rest automatically 
in a ttansverse dneetton over the eauiacie itself foi suifictm>. 

Finally, the* leadscrew spindle, calkd for short the " leadseicvv," 
wns <o arranged that by a set of gears of various diameters, 
a variable, but at the same time foi each train of gcnn* 
fixed ratio between the number of revolutions of the head- 
stock, I.G, the workpiecc, and the loadacrow was obtainable, 
thus making it possible to cut different pitches of threads on 
the lathe. Fig, 2 given the general arrangement of such a 
lathe. * 

B 2 

4 The Colt illation of Changc-irhceh 

The leadscrew revolves m the leudseiew-nul, which is 
fixed to the apion, and, as this nut cannot revolve, it travels 
along the leadscrew, the caniage at the same time maKtn a 
coi responding movement 

The movement of the carnage aheady causes a ronstdeiahle 
piessuic on the thread of the leadsciew and the nut, whit h 
is still inci eased by the cutting of the tool on the matenal, 
and, as a natural icsult, both the leads<iew and the nut aie 
exposed to a ceitam amount oi wear. 'I his weai i\ ttnther 
mci eased by swaif and chips falling on the le.ulserew, and their 
getting between the nut and thiead, 

It is evident, as far as the leadsciew is concerned, that 
this wear will only affect that poition ovei which the mi! 
travels on the thiead. As the woik on the Kit lie v,uies HI 
length (but is as a inlc considerably shoitei than the m.iximwn 
distance between the centies), the wear of the tluead is 
gieatest on those parts of the leadsciew where the nut mv<*s ( 
and after being in use foi a certain time, it is impossible to 
pi event the leadsciew being scarcely worn at all at the end 
but considerably woi n in the centre, and worn most f nil 
close to the headstock, The wear of the nut, however, is 
fairly even. 

The nut was foimeily made solid, consequently it was 
impossible to icpair the wear. It wus soon seen, however, 
that it was piefeiablc to have half nuts, .so that not only can 
it now be repaired, but, by means of the lever a (Fig, 2), it 
can also be opened and closed. 

This has led to the attainment of a number of advantages 
First and foiemost, the possibility of repairing the nut just 
referred to A downward pressure of the level a keeps both 
halves of the nut closed so as to grip the losukserew. The 
two halves of the nut hb move in a vertical direction at the* 
back of the piece c, and are provided with pins which fit in 
cccentiic slots in the citcular plate which revolves on point / 
Fig. 3 shows these cccentiic grooves in the plate, If the pins 
of the half nuts aie shifted by moving the levi ,/, the half 
nuts travel the double distance A B (Fig, 3), vk, : th upjnir 
nut up and the lower one down, the half nuts being thus 

jor Screw-cutting on Lathes 

entirely disengaged from the thiead, causing the motion 
imparted to the carnage by the Icadscrew to cease im- 

In the eailier types of consli action, with the solid nut, the 
carnage had to be moved by hand by means of a handle 
placed on a spindle in the apron, with a bevel gear on the 
other side of the spindle to which this handle was attached , 
this in its turn meshed with anothei bevel geai fixed on the 
hub of the nut. In this way the 
nut was made to i evolve ovei the 
leaclsciew and the cainage was 
moved over the bed. But it took 
far too long to move the cainage 
any distance at all ovei the bed, 
besides being very fatiguing work 
The nut, being in halves, can no 
longei revolve, but it can be 
opened. A lack is to be found on 
the bide of the bed in which a 
pinion meshes to which motion is 
impaited by the hand wheel h 
(Fig, 2), by means of which the 
caniage can be quickly disengaged 
fiom the leadsciew, and a quick 
and easy hand movement is secured. 

Other advantages besides those 
enumeiated here have been clenved 
from the split nut. One gieat 
difficulty, however, still remains, 

viz,, the diffeient weai on a ceitam length of the leadscrew If 
this happens to be more worn in the middle than at the ends, 
it is impossible to cut a true thicacl 

Now, in comparison to the work oidmaiily pei formed on 
a lathe, but little screw-cutting is done. The greater part of 
the time the leadscrew is thus engaged foi the feed motion of 
the caniage and for surfacing* For this reason, the movement 
imparted to the carnage for screw-cutting, has been separated 
from that for feed motion, A separate shaft, provided with a 


6 The Calculation of Change- Wheels 

for St.rew-cit,tting on Lathes. 


keyway, impaits motion to the pinion which meshes with the 
lack (Fig 4), by means of bevel and spui-geais The sliding 
movement of the caniage being accomplished in this mannei, 
the leadscrew is only used foi screw-cutting In still latei,and 
principally Amencan constiuctions, the two shafts have finally 
been united in one, the leadscrew being now piovicled with a 
keyway, for sliding and sui facing the leadscicw simply acts 
as driving shaft, the thread of the leadscrew being only used 
foi sciew-cuttmg, and so the same object is attained with one 
shaft as is obtained in Fig 4 with two, viz., the thiead of the 
leaclsciew is used for sciew-cutting only 

FIG. 5 

In Fig, 2 the gearing for the motion of the leaclsciew from 
the head spindle is cleaily visible. Wheel I is keyed to the 
head spindle ; icar wheels 2 and 3 mn loose on studs fastened 
to the levei 4 By means of knob 8, this lever can be laised 
to hole 5 or lowciecl to hole 6, If the lever is placed in 
position 5, wheels 3 and I become engaged, and wheel 10 on 
spindle 7 revolves by means of wheel 9, Wheel 2 now mns 
to no pin pose. If the lever is placed in position 6, wheels 
2 and i become engaged, and wheel 3 is brought into play by 
means of wheel 2, thus causing wheel 3, as well ab wheel 9 and 

8 The Calculation of Change-Wheels 

spindle 7 to rotate in an opposite direction In the illusli.i- 
tion the levei stands midway, so that wheel I engages neither 
of the wheels 2 or 3, consequently, although the lathe spnulk* 
lotates, the leadsciew is not rotating Wheels I, 2, 3 ,unl <j 
have the same numbei of teeth, so that the wheels on spindle 7 
make precisely the same numbei of revolutions as the lathe 
spindle. Wheels 10, 1 1, 12 and 13 are the actual chan&e-wlu'c'ls, 
and can be easily mounted, dismounted, 01 changed Wheels 
ii and 12 rotate on a sleeve on spindle 14, and consequently 
make the same number of i evolutions, so that wheel \?, 
transmits veiy slowly to wheel 13 the motion hnpaih'd to 
wheel ii In the illustration the gearing between wheel 9 to 
the leadscrew is accomplished by 4 wheels wheels 10 and 12 
being the driving wheels, ii and 13 those cluven It is evi- 
dent that the motion of wheel 9 on spindle 7 it. mipaited 
but veiy slowly to the leadsciew, in the same iatio as the 

product of the number of teeth on wheels 10 and 12 to the 
number of teeth on n and 13 Precisely the same is to be 
seen in Fig 4. Wheel 13 can, however, be driven by means 
of a wheel engaging both wheels 10 and 13, without the 
mteimediate wheels ii and 12, thus solving a,s an idle wheel, 
m which case wheel 10 is the chiving wheel and 13 the 
one driven The iatio between the numbei of icvolutuws ol 
the lathe-spindle and leadsciew is identical with the nttm 
between the numbei of teeth on wheels 10 and 13. 

Wheels n and 12 are mounted on a sleeve iimnint' mi 
stud 14 (See Fig. 6) 

This stud must be movable in accoulauco with the 
dimensions of the wheels, and is consequently placed in 
casting called the sheai 01 swingplatc at the end of the lathe 
This shear (Fig. 7), has two long slots, so that the stud can 
either be brought close to the leadscrew B, for small wheels, 

for Screw-tutting on Lathes. 9 

or moie to the icai foi latgei wheels, at will In ordei to pei- 
mil of working with five or six wheels, <i second slot is to be 
found in the sheai This shcai turns on the le.adgc.rew B, 
and is held in position by means of the two bolts to be' seen 
in the cucular slots When the mtei mediate wheels have been 
accuiately set in the wheel on the leadscrew, the shcai, which 
was first loweied to its full extent, is laiscd till the mtci- 
mediate wheel engages the uppei wheel piopcily, aftci which 
the shear is fastened. 

Fig, 5 shows an Amencan type of lathe, on which it is 
not necessaiy to change the wheels for diffeient pitches of 

FIG. 7 

Lhieads By means of a conc-geat to be found under the 
headstock and at the left-hand side of same, the latio of speed 
between the lathe-spindle and the leadsciew can be varied 
by the simple movement of a lever. The necessity of calcu- 
lating the change-wheels is done away with, all that is icquiicd 
being the placing of two levels in a ceitam position indicated 
in the table. The manner in which this result is attained will 
be {wilier described in Chapttte HI, 

,* ] 

io The Calculation of Change- Wheels, 



(a) System* 

IN the calculation of change-wheels foi sciew-tuthng on the 
lathe there is one difficulty, and that is, the diifeience between 
the English and metric system of measuicmenLs. It is not 
Insurmountable, but it does not icnclei the task any easiei, 
and has been the cause of a consideiable amount of tiouble. 

In the calculation of change-wheels it is <i mattei of in- 
difference whether a light- 01 left-handed sctew is to be cut, 
what form the thiead has to take, whethei the thiead is 
internal 01 external, 01, finally, the exact internal 01 external 
diameter of the thread The one essential question to be 
answered is How many thi cads, aic icquited foi a ceitain 
unit of length ? 

For this puipose two units exist , ist, the inch ; 2nd, the 

Foi both these units of length the numbei of Kvolulions 
of the thread are tcimed "numbei of thiedds." 

The length of a single thread is spoken of as " pitch." 

The number of thread* is thus determined dy the numbet of 
revolutions per ttmt of length 

If the pitch is indicated with the inch as the unit of length 
we speak of " English thiead." If the pitch is indicated with 
the centimetre as unit of length, it is culled a " metnc thiead " 
Both, however, have a system, which rs further treated of in 
Chapter III, but which, as such, has nothing a t all to do with 
the calculation of the change-wheels. , 

If but one of these two units, cither the inch or the centi- 
metie were exclusively adopted as the stunclau! unit, then the 
difficulty refeired to at the beginning of this chaptei would 

for Strew- cutting on Lathes 1 1 

entirely disappeai. But the inch and the ccntimetie aie em- 
ployed together , and not only that, but theie is also a lack of 
unifoimity with legatd to the leadscrcw, one makei cutting 
the leadscrew according to the English, and another accord- 
ing to the metric system. English and American lathes 
usually have a leadsci ew cut accoidmg to the English system , 
Ficnch and Swiss makers cut it almost exclusively accoidmg 
to the metnc system, whilst Geiman manufacttueis employ 
both systems, though the pieference is given to the English. 
Foiu variations aie thus possible . 

I. A metric thread to be cut on a lathe with metiic 

; 2. An English thiead to be cut on a lathe with 

English leadsci ew 
3. An English thread to be cut on a lathe with 

metnc leadsctew 

4 A metric thiead to be cut on a lathe with English 
leadsci ew 

Buefly summarized 

To cut i. Metnc on metiic 

2 English on English 

3 English on metric 
4, Metric on English. 

If one desnes, once and for all, to be able to calculate 
the change-wheels loi eveiy vanety of pitch, it is impel alive 
to know these four varieties thoioughly, as they can occur 

1st Axiom, -The number of tlircadt, ts to be determined by 
the pitdi of the leather ew and the uilio of the number of revo- 
lutions of the laths spindle to that of (he Icadscn'io 

This axiom holds good for all four UZ.WM. 

The ratio of the number of revolutions of the lathe-spindle 
to that of the load-screw is obtained by means of wheels 

When the spindle of the lathe has completed one revolu- 
tion, then the woik on the lathe will have also completed 
one revolution. 

1 2 The Calculation of Change- Wheels 

If the number of i evolutions of the lathe-spindle <uul lead- 
sciew rue the same, so that the leadsciew has also completed 
one revolution, then the camai>e has moved a distance dining 
this one i evolution equivalent to one thicad of the leadsctew 
If a tool has been placed in the toolholdei, so that it can cut 
the woik-piecc, then piecisely the ime pitch will have been 
cut on the woik-picce as that on the leadsciew. With an 
equal number of revolutions of the lathe-spindle and the lead- 
screw, the tin cad cut on the work-piece will have the same 
pitch as the leadsciew 

If the lathe-spindle has completed one full i evolution, 
but the leadsciew on the other hand only half a i evolution, 
then the carnage, and with it the tool, will have moved in a 
straight line ovei a length equal to half a pitch of the 
leadsciew It is thus only when the lathe-sptndle has 
made two i evolutions that the leadscrew will have completed 
one full i evolution, two threads aie now to be found 
on the work-piece over a length equal to one pitch of the 
leadscrew. The ratio of the numbei of i evolutions of the 
spindle to that of the leadscrew was 2 I , the latio of the 
number of thieads per unit of length of the woik-piece to 
that of the leadscrew was also 2 i. Hence it follows 

2nd Axiom, The ratio of the number oj i evolutions of the 
lathe-spindle to that of the leadscrew u the same as the pro- 
portion of the pitch of the thread to be cut to that oj the lead- 
it crew 

Axiom 2 is also applicable to all foin cases. 

Foi example, the leadscrew of a lathe has a pitch of one 
thread to the inch It 14, lequiied to cut two threads to the 
inch. The piopoition of the pitch to be cut to that of the 
leadscrew is thus 2 i Accoiding to axiom 2 the ratio 
of the number of revolutions of the lathe-spindle to that of 
the leadscrew must also be 2 I 

The leadscrew has thus to complete one i evolution to 
two of the lathe-spindle. The leadscrew leceives its motion 
fiom the lathe-spindle, so that the rotation of the leadsciuw 
must be ictarded accoidmgly. The lotation of the lathe- 
spindle is transmitted to the leadsciew by wheels. The pro- 

for Screw-cutting on Lathes. 13 

portion of the numbei of teeth on wheel 10 (see Fig. 2), to 
those on wheel 13 on the leadscrew must thus be in inveise 
piopoition to the latio between the number of revolutions of 
the lathe-spindle and the leadscrew, which, in the example 
given, must be 2 1 , the latio of the wheels 10 and 13 thus 
becomes I 2. If then a wheel with 50 teeth be on the 
sleeve of spindle 7, and one with 100 teeth on the Icadsciew, 
with any desued idle wheel, a sciew of 2 tin cads to the inch 
or }-mch pitch will be obtained on the work-piece with a 
Icadsciew having one-inch pitch. From this we amve at 
what is again applicable to all foui cases 

jrd Axiom The- proportion of the number of the threads to 
he cut to those in the leadscrew is in inverse ? atio to the pro- 
portion of the number of teeth on the wheel on the lathe-spindle 
to the number of teeth on the wheel of the lead-screw, 01 In 
fiactional foim 

Number of thicacls to be cut _ 
Numbei of tin cads in the leadscrew ~~ 

No. of teeth on the leadscrew wheel 
No. of teeth on the lathe-spindle wheel 

In this manner the calculation of the change-wheels foi 
scicw-cuttmg is i educed to the woikmg out of a simple 
fi action the number of threads to be cut being the 
numerator, those in the leaclsciew being the denominator, or, 
if it is denned to expiess the fraction in the same manner as 
the wheels, i e. the numbei of teeth on the lathe-spindle wheel 
on top as numeratoi, that of the Icadsciew underneath as 
denominatoi, it is just the rcvcisc. The number of threads 
in the leadscrew will then represent the value of the 
numerator, those of the thread to be cut icpiesenting the 
denominator As the pitch of the leadscrew on a ceitar 
lathe is always the same, it follows that the value of *' 
numeratoi is always constant, 

We must here call especial attention to a rmsundcrsta 
which so often occurs m connection with the question 
whether the number of threads in the leadscrew must 

14 The Calculation of Change-Wheels 

the numeratoi or the denominate! A practical man can 
generally tell fairly well which wheels have to be placed on 
top and which underneath, but still, when the pitch of the 
thiead to be cut closely appioximates that of the leadsciew, 
mistakes can sometimes be made 

The screw may be denoted by the numbci of thieads pet 
unit of length, m which case the numbei of thieads in the 
leadscrew is the numerator of the fi action 

The screw may also be denoted by the length of one pitch 
of the screw , m this case the length of pitch of the scicw to 
be cut will be the numeiatoi, the length of pitch of the lead- 
screw being the denominator of the fiaction, the numeiatoi of 
which will indicate the numbei of teeth on the lathe-spindle 
wheel, the denominator indicating the number of teeth of the 
wheel on the leadsci ew 

Should the numbei of thieads of the sciew to be cut be a 
multiple of those m the leadscrew, one is naturally inclined to 
express it in number of thieads per unit , foi example, 4 
threads per inch to be cut on a lathe with a leadscrew of 
i thread per inch , should it not be a multiple, as for 
example, each thread having a length of 7 mm., one is then 
inclined to denote it by the pitch. If, m both instances, the 
number of threads in the leadsciew be r pci inch, the fiaction 
in the first instance will be 

Number of thieads in the leadscrew _ , _ chiving wheel 
Number of threads to be cut wheel to be d liven 

In the second instance, in which the pitch of the screw to 
be cut must be 7 mm., the number of the threads to be cut 

per unit is itself a fraction, viz - 5 4 , the fraction thus being 

i 7 

. 7 being the length in mm. of the pitch of the 

fo _4 ^3 4 

7 ~ 

screw to be cut, 25 4 the length in mm. of the pitch of the 
lead-screw, so that, in this case, the length of pitch of the 
screw to be cut can at once be placed m the numeiator foi 
the driving wheel, the length of pitch of the leadscrew being 

for Screw-cvitttng on Lathes 15 

placed in the denommatoi foi the wheel to be duven In 
actual calculation the fotegomg examples must be carefully 
distinguished one from the othei 

(b) What Change-wheeh aic to be found on a Lathe 

This question piescnts itself each time change- wheels have 
to be calculated, because the fraction which is formed by the 
thiead to be cut and the leadsciew, must be changed into one 
foimed from the wheels to be found on the lathe These 
wheels should have such a number of teeth as will, within 
ceitam limits, include the indivisible factors, viz 2, 3, 5, 7, 
II, 13, 17, 19, 23, etc Some makeis supply these wheels in 
a piogiession of 5, othei s with another progression. The 
following set of change- wheels is, or should be piovided with 
eveiy lathe, vu 

15 = 3x5 60 = 2x2x3x5 

20 = 2x2x5 65 = 5x13 

25 - 5 x 5 75 = 3 x 5 x 5 
30 = 2 x 3 x 5 85 = 5 x 17 

35 = 5 x 7 95 = 5 x 19 

40 = 2x2x2x5 100 = 2x2x5x5 
45 = 3 X 3 X 5 105 = 3 x 5 X 7 

50 = 2x5x5 115 = 5x23 

55 = 5 x ii _ 125 = 5 x 5 x 5 

If) = 2X2X2X2 42 = 2X3X7 

18 = 2 x 3 x 3 44 SB 4 X ii 

20 = 2x2x5 56 = 2x2x2x7 

21 = 3x7 60 =2x2x3x5 

22 = 2XH 66 = 2X3X11 

26 = 2x13 78 = 2x3x13 

28 = 2X2X7 88 = 2X2X2X11 

34 = 2x17 96 = 2x2x2x2x2x3 
38 = 2x19 1 08 = 2x2x3x3x3 

One of the two foiegoing sets is gencially provided with 

1 6 The Calculation of Change-Wheels 

the lathe English lathes usually have a set of 22 wheels 
some of which have the same number of teeth 

It will be clear from what has been said, thus far, that 
the easiest thiead to be cut on a lathe, i e the thread causing 
the least tiouble m the calculation of the change-wheels, is 
that having the same system as the leadsciew This will be 
the case with the ist and 2nd cases refeired to on page II 

(c) The Cutting of Metric Threads on a Lathe with Metric 


Take the case of a lathe with a leadscrew having I cm. 
(10 mm ) pitch. 

It is requiied to cut 4 thieads per cm. 

No of teeth on driving wheel _ No of thieads in the leadsuew 
No~o7teeth"oia wheel to be driven ~~ No, of thieads to be cut 

= 1 = _*L = gear-wheel 10 1 Sce F]g 2 

4 100 = geai -wheel ou lead-screw J 

It is required to cut 7 thieads per cm. 

No of thieads in the leadscrew _ I _ 15 = driving wheel. 
No of threads to be cut 7 105 = wheel to be duven 

To cut i^ thread per cm. 

No of threads m the leadscrew ,_ j_ _ 5 or 6 = cluvmg wheel. 
~No of threads to be cut 15 75 go = wheel to be duven. 

To cut 3 threads per cm 

No of threads in the_leadsciew I _ 30 = driving wheel 
No of threads to be cut 3 90 = wheel to be driven 

To cut 5 threads pei cm. 

No of threads in_the_leadscrew I 25 = driving wheel. 

No of thieads to be cut 5 125 = wheel to be duven. 

In the last example it is also possible to say, a pitch of 
2 mm , in which case the fraction will be 

Pitch in mm to be cut _ 2 _ ^25 = duving wheel 

Pitch in mm of leadscrew lo 125 = wheel to be driven. 

In both cases the result will natmally be the same. 

Jor Scrcw-cnttuio on Lathes. 17 

Pitch in mm to lie cut _ 7 _ 70 = duving wheel 
I'Ucli in mm of leadseiew 10 100 = whcc>l to he duven 

Pilch in mm to ho tut _ 5 5 _ S5 = dnvm^ wheel 
Pilch in mm ol k'idseiew 10 100 ~- wluel to In duvcti 

To cut a pitch of 7 mm 

Pitch in mm to be cut 
Pitch in mm of leadseiew 

To cut a pitch of 5^ mm 

Pilch in mm to ho tut 
Pilch in mm ol k'idseiew 

To cut 7 threads pei 22 mm 


Denoted in pitch = a pitch of " mm 

Pitch m mm to he cut _ "/- _ 22 ~ dnvinu; wheel 
Pitch m mm ol leadsuew TO 7 " wheel to lie duven 




Fu>. 8 Fiu. 9. 

So fai it has always been possible to work with a single 
tiam of wheels with any desued idle wheel. Fig, 8 shows a 

single train. 

1 8 The Calculation of Change-Wheels 

In the set of wheels to be found on the Lithe, wheels with 
cithei 22 or 70 teeth, as piesumed weie employed foi the 
pieccding examples, weie not included A compound tiain 
is now used. 

22__2XH _ 20x55 dnvtntf \\liccls 

70 ""7x10 ~ 35X HID wheels lo lie dnvcn 

Fig 9 shows this comj)ound tiain 

a and /; are the drivers, c and //those d) iwu The fixing 
up of the wheels will thus be 

55 x 20 
100 x 35" 

The wheels m the numeiatot, as well as those in the 
denominate!, can be interchanged , a may thus be put in 
place of b, 01 c in pLicc of </, or both may be c lumped , btit 
interchanging of a clnvei wilh one to be diiven may nevoi 
take place, as this would altei the value of the fraction and 
an entirely different thiead would be obtained. 

It is always advisable to tiy to get the smallest of the 
drivers on the lathe-spmdle, and the laigcst to be driven on 
the leadscrew, in oidci to obtain as lational a geating as 

To cut 1 1 threads pei 14 mm I he pitch is thus 14/11 mm, 

Pitch on leadsciew ro mm. 


2X / 

ro n x 10 j i x 10 55 x 100 

To cut 3^ thread pci 40 mm. The pitch is thus 40/3 * 5 

Solution 40/3" 5 = 40 ^ 4 X 10 ^ sox IOO ( 
10 3^x10 SX; 5^X35 

To cut /] tin cads on 15 mm The pitch is thus 15/4 nun. 

IS ^3X5 ^30x50 f 
10 4 x 10 4x10 40 x ioo 

Should the lathe have another pitch than 1 cm., this will 
only necessitate a change m the constant of the leadsercw In 
the fi action. 

for Screiv-cutting on Lathes. 19 

The following aie a few examples with solutions, dealing 
with diffeicnt leadsctcws 

To cut 9 thieads pei 16 mm , leadsciew 2 threads per 
i cm The pitch of the thiead to be cut in 16/9 mm 
The pitch of the leadsciew is 5 mm. 

c , , 1 6/9 2x8 20 X 40 . . , , 

Solution ' - = ^ = in case these wheels 

5 9X5 45X50 

, n 20x80 

are too small 

45 X 100 

To cut a pitch of 3 mm. Pitch of leadsciew being- 7 5 mm. 

2 -2Q 

Solution 7 f 5 = ^ 

To cut 8 threads pei 13 mm Pitch of Icadscrcw, 7" 5 mm 

Solution 13/5 '3 =2x6 5^20x65 

75 8x75 8x7-5 80 x 75 

In both the foicgomg examples, a wheel with 75 teeth 
appeals among the wheels cliivcn, but is not included in the 
specification given on page 15. With a leadsciew having a 
pitch of 7' 5 mm. a wheel with 75 teeth will icpcatcdly 
occui , in such a case the manufactiuci will be certain to 
supply a wheel with 75 teeth. 

To cut a pitch ot 20 mm Lead screw pitch 25 mm 

c . 20 100 

Solution 25 = 125' 

To cut 3 threads per 20 mm. Leadsciew pitch 25 mm. 

t , . , 2O/3 2O 2 X 10 40 X 50 

solution f.^i ~ ~ 

solution. 25 - 3 x 35 - 3 x as - fo x I2S - 

To cut a pitch of 37-5 mm. Leadsciew pitch 25 mm. 

Solution. 37-5^15X25 30x100 
on. 25 10 X 25 40 X 50 

To cut a pitch of 76 mm. Leadscrew pitch 25 mm. 
76 4 x 19 40 x 95 80 x 95 

Solution _ 

35 2'SXIO 25X50 25X100 

C 2 

2O The Calculation of Change- Whech 

(d] The Cutting of English Threads on a Lat/ii unfit 
English Lcadscrctv 

In principle, this second case lescmblcs the fiist The 
system of the leadsciew and the Ihicad to be cut is the 

Most lathes have a leadsciew with ^ in pitch, thus 2 
threads pei inch Heavy lathes have a leadsciew with i in 
pitch, the smallei sizes in, or 4 thicacls per Inch, whilst in 
exceptional cases 2\ thieads pei inch aie to be found. Given 
a ceitain pitch, the ft action can then be clctci mined without 
any difficulty 

Should the screw be denoted in a ceitain numbei of 
thieads per inch, the number of tin cads pei inch of the lead- 
sciew is placed in the numerator, the nunibci of thieads pei 
inch to be cut in the dcnorninatoi Should the sciew be 
denoted in the length of the pitch, then the length m inches 
of the pitch to be cut is placed in the numeiator, the length m 
inches of the pitch of the leadsciew being placed in the 

In practice the majority of threads aie cut accoi cling to the 
Whitwoith system (see page 57), for which reason we shall 
fust of all give a number of problems with solutions fot this 

To cut I in Whitworth thread Leadsciew 2 thieads per 
inch J in Whitwoith thread = 16 thieadb per inch. 

Solution No of threads in_le_adscrew P C1 " inch 

No of thieads to be cut pei inch 

= 2 = _?! *_L _ _ 2 1 * S 
16 16 x 12-5 ~~ 80 x 125' 

To cut ji in Whitwoith thicad Leadsciew 2 thieads 
per inch i\ m Whitworth thread = 7 threads pei inch. 

Solution 2 = - 2 -*_L 2 .X. 40 

7 3*5X2 35 x 80 

for Screw-t,nttmg on Lather. 21 

To cut 2 in Whitwoith tin cad Leadsaew 3 threads pei 
inch 2 in Whitwoith tin cad = ^ thiead pei inch 

c . 4 2 2 X IO 40 X 50 

Solution: = = J 

45 5X9 45 X 100 

To cut 3 in Whitwoith thiead Leadscu-w 2 tin cads pet 
inch. 3 in Whitwoith thiead = 3,\ thiead pel inch 

c , 2 40 

bolution : = - . 

3'5 7 

To cut ij in Whitwoith thiead Leadsaew ,j tlueads 
per inch I \ in, Whitwoith thiead = 7 tin cads pei inch. 

Solution = 

7 70 

To cut i^j in. i>as tin cad. Loadsuew 2', thiead pel inch- 
1 1 in jjas thiead = II tin cads per inch 

,, , t 2\ 2S 20 X JO 

Solution - = = * . 

ir no 5 5 x 80 

To cut 2]- thiead pei inch, Loadsctcw 2 tin cads pel 

c , . 2 2X4 2 X 'I 20 X ()0 

Solution - r =- = ' = 

2 ! 9 3X3 3^ X 15 

To cut I thiead pei inch (not it 1 indi pih //), Leatlsciew 
3 tin cads per inch 

c , ., 2 2 X H 80 .10 X tOO 

Solution w sss s=s ss. 

H 7 35 25 x 70 
To cut 2\ thiead pel inch. Londscicw 2^ thiead per inch. 

Solution; 3 i aa ]ro sa| 50 i 

2% n 55 

In the following examples, the length of pitch is given. 
The pitch of the leadsciew will consequently appeal in the 

22 The Calculation oj Change- IV liceL 

To cut a j in. pitch Leadscicw 2 thicads pei inch =t 
^ in. pitch. 

Solution I^H^fo' 

1o cut a | in. pitch. Lcadbcrevv 2^ tin cads pel inch 

a= r in. pitch 

c . . }-?, }f. x 2-V it; x i\ 75 x icx) 
Solution -- = ' (> = > ~ = x J ., . 

i i 16 x i 40 x 80 


To cut 19 threads on 11-5 in Leadsciew 2J, tlneads per 

inch. The pitch to be cut = in. The leadsciew 


pitch ~ in 


Solution -I2_= !I 5x2.5^115x1125. 

_ 19 95 x 100 


To cut a pitch of 4 % in. Leadsciew I pitch per inch. 

Solution: l5 f = 3 X 13 = 6 5 X 75. 
i 8 2x4 25 x 40 

To cut a i J in pitch Leadsciew 2 thicads pci inch. 
Solution il = I 3X2 == 2x I 3 _65 t 

i- 32 2 X 1C 80 

To cut 9 threads per 5^ m. Lcadsciew 2\ tin cad per 
h The pitch 

leadscrew = -v in. 

inch The pitch to be cut = ^ in The pitch of the 

d La 

Solution JL _ 5if! X 2& ^ 95 x_2/5 125 v 95 
_i 9 9 X 16 80 x 90 " 


In the foregoing examples practically every case which Is 
likely to occur, has been treated. 

for Screw-cutting on Lathes. 23 

(a) To cut English Threads on a Lathe with 
Metric Leaded evv 

In the first and second cases considered, the system of 
the thread to be cut and that of the leadsciew weie identical, 
viz., in the fust case accoidmg to metnc measurement, in the 
second, according to the English measuiement 

In the thud case, howevei, the system of the thread to be 
cut and that of the leadsciew aie dissnnilai The leadscrcw 
is divided per cm = 10 mm., or some pait 01 multiple 
theieof, the screw to be cut being divided pei inch = 25 4 
mm., 01 some part 01 multiple theieof 

In the third case to be considered, this numbei 25-4 will 
consequently appeal icgulaily eithei in tne numciatoi 01 
the dcnommatoi, and will mvauably produce a fi action 
which, with one exception, cannot be lesolved into whole 

An equivalent must theiefore be found, by means of which 
it will be possible to foim a divisible numbei from the nume- 
rator and denominator of the fraction 

This equivalent is to be found as follows 6^ in. = 
16-509675 cm., taking for gi anted that 6\ in. = 16-5 cm., 
there is then a disci epancy of 0-09675 mm. per 165 mm. 
of length, 01 rathei less than 0-06 pei cent., a diffeience of 
piactically no mipoi Lance whatcvei. 

If the number of tin cads to be cut be expressed in a 
certain number pei 6 5 in , and the number of threads of the 
leadsciew be also expiessed in a certain numbei per 6-5 in. 
or 10 '5 cm., the result will be an equivalent which can be 
made use of. 

As reference is heie made to a ceitain numbei of thicads 
per unit of length, in this case, 6-5 in. 01 16-5 cm., the numbei s 
of thicads of the" leadscrcw will appeal m the numeratoi, the 
number of thieads to be cut in the denominator. 

The following comparison can thus be formulated 

No of threads in leadsciew per 16" 5 cm, ^ chiveis 

No. of threads to be cut per 6-5 in, wheels to be driven 

24 The Calculation oj Chaugc-lVhccL 

As the numbei of thicads in the leadscicw lenuuns in- 
variable for the same lathe, the numcuitoi is consequently a 
constant factor foi a ceitain lathe, 

Should the leadsciew have a I cm pitch, the hvulsciew 
will then have 16-5 thieads pci 16 5 cm , and the constant 
factor of the numciatoi will be 16*5, whilst, at the same time, 
6* 5 is to be found as a constant factoi in the denominatoi, 
and must constantly be multiplied by the number which 
expi esses the number of threads to be cut pei inch I( both 
these constant factois be multiplied by 10, the numbei 105 
will always appeal in the numciatoi and the numbei 05 m 
the denominator, in this way 

constant factoi of numerator 165 

denominator No of thieads to be cut pei in X 65 

or .. ll x J 5 

threads pei m X 65 

The equivalent is now complete , by icplncnu* threads pet 
znckm the denominatoi by the actual numbei, a fi action is 
obtained which will permit of the calculation of the wheels, 

In the examples which follow, eveiy possible vaiiation 
has been carefully worked out, fiom the simplest to the most 

To cut 6 threads pei inch. Leadsciew 10 mm pitch, 
Numeiatoi = _ 0x15 11x15 

Denominator = No of thieads pei inch x 65 ~" 0x65 

= 55X75 = 50X55 
150x65 65x100* 

To cut 4 threads per inch. Leadsciew 10 mm, pitch. 

Solution IIXI 5 = 11x75^55x75 
4x65 20x65 65x100' 

To cut 2 7 | tin cads pei inch Leadsciew 10 mm, pitch. 

Solution IIX 15 = 4Xnxi5 = 2X2X 3x5x11 
4X65 9x65 3X3X5X13 

-, I2X 55 ^ 55x6o 
45X13 45x65* 

for Screw-cutting on Lat/tes. 25 

To cut 5\ tin cads per inch Lcaclsciew 10 mm pitch. 

c 1 4 11x15 1 1 X 30 30 

Solution ; r = ~. = , 

5 5x65 11x65 65 

To cut I in Whitvvoith-thiead = 8 thieadb per inch Lead- 
bcicw 5 mm pitch. 

In this case the Icadsciew has 2 tin cads per cm Conse- 
quently for this p<irticulai lathe, the numeiatoi ib 2 x 165 
= 330 01 II X30 

Solution 11X30^60X55 

8 X 65 65 X So 

To cut } in yat, thieatl = I4thiea ( ds per inch Lcadscrcw 
5 mm. pitch 

014. ii xso 30 X 55 

Solution ~ = , J . 

14x65 65x70 

To cut -]_ in. Whitwoith -thi cad = 20 tin cads per inch. 
Leadscievv 5 mm. pitch, 

c 1 4* II X3O 30X55 

Solution J = ; " . 

20 x 15 65 X 100 

To cut I in. gas thiead = II thieadspei inch Leadsciew 
(> mm. pitch No. of threads in leadscrew per cm., ^P. 

Solution " ' SAA " t3 = " ; i5 =-7 V 5 =r 5 - 
11x05 65 6x65 05 

To cut 36 tin cuds pei inch. Leadscrew 4 mm pitch. 
No. of threads in the leadsciew per cm , l t 01 2*5. 

Solutmn. -5xnxi5 1 .nx-5_ 35X55. 

36X05 I ' X' 3 ' T X 120 

To cut I thread per inch Leadscrew 10 mm. pitch. 

IT x 120 _ 55><i7o 

(""* -I . JU J /\ IT ^' "n * * <\ * 

boluuon. == 7x( . 5 _ 7xfl5 - J5X65 

26 The Calculation of Change-Wheels 

To cut a in pitch Leadsciew 10 mm pitch. No. of 

threads per inch = = . 


8 / 

Solution * = ' -* = ^ 

8 - 8x65 65x80 


To cut 3 thieads pel 2 in. Leadscrew 6 mm. pitch 
No of threads pei inch i| No. oi thieads in the leadsciew 
per cm \ 

c 1 4. V x ii x 15 10 x ii x 5 55 x 100 
Solution . (> , . -*= 3 = i- 3 

J x 65 3 x 65 30 x 65 

To cut 36 threads per 7 in. Leadscrew, 7 mm pitch. 
No. of thieads per inch ty No of threads in the lead.screw 
per cm \ Q . 

Solution- Vxiixis^ioxii Xi S 

a 7 b X 65 36 X 65 

_ 5^x n _ 50 x 55 
6 x 13 60 x 65" 

To cut 9-5 tin edd per 8 inch. Leadscrew, 10 mm. pitch. 
No of thieads pei inch, ^ 5, 

- TI X 15 ^ 8 X ii x 15 _ no X 120 


To cut 25 thieads pei 3^ in. Leadscrew, 5 mm. pitch. No. of 

threads per inch, |i = Io . No. of thieadb in the leadscrew 

3i 15 
per cm, = 2 

Solution 2 x u x I S = ^ X ii X 15 X 15 
Vy x 65 ioo x 05 

= 5S x (JO . 
65 x 100* 

for Screw-tutting on Lathes. 27 

To cut a 2J> in pilch. Leadsciew, 10 mm. pitch No. of 

I 2 

tin eads pel inch, = 

. , A . 11x15 5x11x15 1 10 x 75 
Solution -, , w = , = / 

^ X 05 2 x 65 20 x 65 

To cut 2 tin cads pet 6,1 in. Leadhtiew, 25 mm pitch 
No, of thieads pei inch, ~ = ^ No, of t hi eads in the lead- 

I 2 

sciew pei cm,, ^f~ * 

" i 5 

t , . ^xiixi 1 ; 2xi>xnxi5 5sx 60 

Solution ' , . J = - 

i x ot; 1 X 5 x 05 k |u x 2t; 

(/) 77^' Cutting (>J Mt'tm Tluead^ on <i Lathe witk 

To some extent the fouith case lesembles the thud The 
piopoition 10:25*4 also holds good, though with an opposite 

Use is also made in this instance of the fact that 6" 5 in. is 
equivalent to 16*5 cin 

Suppose, for example, that the leadscrew has a t inch 
pitch and 10 tin cads per cm. have to be cut, i.e. a I mm. 
pitch, then, when the leadscrew lias completed 6-5 revolu- 
tions, the lathe spindle should have made 165 i evolutions, 
which can be formulated 

No. of threads in the leadscrew pci 6-5 in, _ 6* 5 
No, of threads to be cut pet 165 mm. ~~ 105* 

The numerator of the fraction will thus, for a yivcn lathe, 
always be equivalent to the number of threads per Inch in 
the leadscrew X the factor 6*5; the denominatoi being 
equivalent to a fraction, the numeratoi of which is the factor 
165, and the length in mm. of the thread to be cut, the 

28 The Calculation of Change-Wheel \ 

Foi example, a 2 mm. pilch is to be cut on a lath 
having a leadscrew of 2 threads pei inch, then 
the numeiator will be 2x6 5 = 13 

and the denominator will be 


Foi this paiticulai lathe the numeiator will always be 13 
The fust lesolvent of the fraction is a whole mmibc 
obtained fiom the clenominatoi by placing the denominate) 
of the fi action, winch is the denominator of the compouru 


fraction in the numeiator, thus r 

No useful puipose, howevei, is effected by this alteratioi 
eveiy time The pitch of the thread to be cut is according!) 
placed directly in the numeratoi, the fraction then benu 
definitely foimulated as follows 

Numeratoi = Pitch m mm, of thiead to be cut x No. of 

thieads in the leadscrew pei inch X 6 5 
Denominator = 165 

Attention must here be directed to the fact that whenevci 
the length of the thread to be cut is a fraction, it must nevei 
be lesolvcd into a decimal, but must always be placed in the 
numerator as a vulgai fraction, so that compound fraction 1 - 
maybe icsolvable from numeiatoi and denominator by multi- 
plication of both. 

The following examples, from the simplest to the most 
complicated, will make cleai what has been stated above . 

To cut a bciew of 5 thieads pei cm Leadscrew 2 threads 
pei inch 

To be cut a 2 mm pitch. 

Solution. 2X?X6- S= 2x13 = 20x65 , 
165 ii x 15 75 x no 

To cut a 3 5 mm. pitch. Leadscrew 2 threads pei inch 

Solution. 3'$xr 3 = 35X6 5> 

11x15 75x110 

for Screw-cutting on Lafhc\. 29 

To cut a sciew of 3 tin cads pet cm, 'Leadseiow 2 thicads 
pci inch , 

To be cut a ^ mm pitch 

Solution ^ X1 3 = 10x13 = 10x1^20x05 
11x15 3x11x15 11x15 ,15x110 

To cut a screw of 8 thicads per I i mm Lc.ulsetcw n 
threads pen inch 

To be cut a y mm pitch 

c , . V x 13 13 20 x 05 

Solution - h = = J . 

11X15 8x15 J GO X \ "M 

To cut a sciew of 5 thieads pei 18 mm Le.ulsdew > 
thicads pci inch 

c i 4 M^xis 13x18 6x13 s <! xfK 

solution __ - - = r- ~~ * " 

11x15 5x11x15 nx^ 55x115 

To cut ,i sciew of 4 thicads pci 7 mm Lisulsucw ?\ 
thicads pci inch 

Solution 1X2^X6^ 7x13 35X^5 . 

11X15 4X2X6X11 I 10 XI 30 

To cut a 7^ mm pitch Lcadsciew 2\ tin ends p<jt inch. 

Solution , 7 ^ x 2 l*fia = 5 X J 3 ^ 5<> X 65 
11x15 tixS 5 5 x tSo 

To cut <i io| mm pitch. Lcadsciew i thuNid pci inch, 

Solution 10 * x(5 i 2IXI 3 ^ 7XI3^3SX65 

LIX15 4XI1XI5 11X30 55x100' 

To cut a 42 mm. pitch Leadncrew i inch pilch. 

Solution 42X6-5 = 42x13 ^ 7X rj ^ 70x65 
11x15 2x11x15 sx n ~ 50x55' 

To cut a sciew of 13 thieads pet 5 mm, Lcadscnnv /j 
thicads pei inch. 

Solution, x/ix 2XS ^ 20x25 

11x15 7-SX22 75x110' 


30 The Calculation of Change- 1 7 '//<'<?/ r ^ 

(g) 7/# Wheel with i 7 /re//' 

In addition to the equivalent 6* 5 in. i<VS < m,, nhirh 
has been employed in the thud <uid fourth cases, theie 
is still another way of cutting English thu-.ul on a lathe 
with metnc leaclsciew, 01 vice vet&d, winch is, by nicking 
use of a wheel with 1 27 teeth 

The pioportion between the cm. and the inch of 10 2}'4 
can be resolved into one of 50 : 127 

127 is not divisible fuilhei, <ind so, if a wheel with 
127 teeth be employed, this i.ictoi can be- placed eilhej in 
the numcratoi 01 the denominate! . 

The thncl and fouith cases will then leseinble the fiint, 
seeing that it is now possible to expiess the Knghsh tlmsul 
in mm, whcthei it be the thieads in the leaclscivw r the 
thieads in the sciew to be cut The ft action will thus be 

Numeratoi = Pitch to be cut in mm 
Denominator = fitch of leadsciew in mm 

Numerator = No of threads in lead screw pet inch 
Denommatoi = No. of threads to be cut pei cm X 2*5-1 

The following examples will cleat ly indicate* what Is 

To cut a 2 mm. pitch. Leadsciew 2 thieads per inch, 
Leadscrew pitch 12*7 mm. 

Numeiator = 2 _ 20 
Denommatoi = 12 7 127 

The foregoing example, when wotked out as per the UiHt 
comparison, will yield the same result, seeing* that : - 

2 mm. = 5 threads pci cm. 

Numeiator = 2 __ 3 __ 20 
Denommatoi = 5x2' 54 ~* 12-7 "~ 127 

for Screw-cutting on Lathes. 31 

To cut 3 threads per cm Leadscrew 2 threads per inch. 

Solution - -- -= ~ 2 ___ = 4X50 = _40>5o_ 

3x254 6x1-27 6x127 60x127 
01, accoidmg to first comparison, 

i pitch = 10 mm. 

Numerator = ip j; O _ 40x50 

Denommatoi = 12 7 ~$xi2 7 Y~~6ox 127' 

To cut 7 thieads pei 44 mm. Leadscrew 2 threads per 


Solution -L_ = _J4_ = 40 x 55 
12 7 7x 12-7 35 x 127 

To cut a 9 mm. pitch Leadscrew 2^ threads pet inch 

Solution -2-.= 9X25 = 45Xi2 S . 
25 '4 254 50x127 

To cut 28 thieads pci 45 mm. Leadscrew 4 thieads per 

Solution. JL= 45X4^45X50 
25-4 28x25-4 70x127 


To cut i m Whitworth-thrcad = 8 threads per inch Lead- 

screw 10 mm Pitch to be cut = ^4. mm. 



Solution: _i_ = 25 '4_ = 20x127, 
10 8xio 10x100 

When cutting metric thread on a lathe with English lead- 
screw, the wheel with 1 27 teeth is always to be found amongst 
the wheels driven, whilst, when cutting English thiead on a 
lathe with metric leadscrcw, it is found among the drivers. 

To cut 3 in Whitworth-thread = 3^ threads per inch. 
Leadscrew 10 mm. pitch. 

c , . 25-4 20x127 

Solution ; - -- J ~ = 

3*5Xio 35x100 

32 TJic Cal( illation of Chtuigc-U7iec/\ 

Yci cut 4 in, <^as thicnd = 11 tlueads pei inch. Leadsnew 
14 > nun. pitch 

e i , 25 4 20 x 1:7 

Solution J h = ' 

I I X 1C) IOC) X I IO 

Io cut 3 tlueads pei 8', ui =- nu h pil<h Lcads(io\v 

o ', 

|l > nun. pitch 


r o 3x10 2 5 x ' 10 

To cut <) tlueads pci 1 1 in Le.ulsuew 25 nun pitch, 

25 A 
1 1 

Solution 9__ ^0x25-4 = 45 xi -7 

25 11X25 55x125' 

To cut 7 tlueads pei 3 in. Leadscrew 7 nun pitch. 

"^^^C'/l ^ O V I ^ 7 

Solution ^ "- 1 ^ = " 

7X7 35X70 

To cut 24 thieads pci c) in. Leadscrew 5 mm. pitch 

c i < 9x215-^ /is x 127 

Solution = 1J , 7 

2/1 X 5 50 X 60 

(//) Melltod for Calculating Approximate F 

Bcfoi e conimcncinjy wit IT, the actual calculation, the question 
was pi opoundcd nuclei heading (//) on p<i t i>e 15 . " What chanp.e- 
whools ue to be found on <i lathe?" This was indeed im~ 
ju'tMtive, as the change-wheels actually pie.sent on the lathe 
have invuiubly to be taken into account, first of all because 
the fraction must be lesolved into munbcis coucspondinjjf to 
the change-wheels, and then, because the same faetois which 
*u to make up the fraction must also be found in the change- 
wheels. Should the fraction contain a fac toi not to be met 
with In the change-wheels, then, aceoiding to the methods 
now in vogue, a suitable set of wheels could not be iound, 

for Sirew-cuttnio on Lathc\, 33 

consequently, the thicad in question could not be cut without 
obtaining one 01 mote \\hecls making up the lequisile faclois, 
which, of coin so, would not be possible, as a certain thiead is 
ijeneially tequired to be cut without notice, and theie is, 
theiefore, no chance of eithei making 01* obtaining suitable 

Will siuh eases often ocem ? Not as a nile The 
examples aheady ^iven cleaily show that even HI the case 
of thieads which vuiy veiy consideiably, the wheels necessaty 
foi cutting ,i line thiead can be found 

In the set of change-wheels, j^iven on page 15, the following 
fact 01 s weie found 2,3, 5,7, n, 13, 17, 19, 23; the factoi 
33 was not met with in the second set, whilst on many lathes 
UK far to is 17, i<), and 23 aie absent 

If jetton ap/hw in //if fnution composed of the thread lo 
fa' cut and the /twhcniv, wJiii/i cannot />e found ut the diau^e- 
w/ft't'/s, then sitth n thififd tannot be ait accurately, 

If it is absolutely necessaiy to cut such a thiead, a ft action 
must be sought lot which appioaches the coirect fraction as 
neaily as possible. 

Lack of knowledge of the conect method of finding out a 
I rarl inn apptoximalimj the tine fi action as closely as possible*, 
loo often tesults in the calculation be-in}; skipped ovei, and a 
li.ution btnn t t chosen which actually ^ives a thiead diffeiint; 
rtnsiderably fioin the one tequired. 

In addition, the fact ts loo often lost si^ht of that an 
approximate fraction will still result in an uuseiviceahk thiead. 

Support", for example, a fi action is found which yields a 
tturad tlifiVrhtjLf only 0*05 mm, fiom the thread of the nut to 
lit which the thread has to be cut. At first si^hl the differ 
cmv appears trifling, but the en or which has been made is 
ieally vwy tfj't'trt, so ^reat, indeed, that the thtead obtained is 
wholly #,w'/d'.vx It must of course not be foi|,;otten that each 
thread inct cases the error, which at the end of 20 tin ends will 
result in a difference of 30 X 0-05 inm, = i mm Suppose, 
fwther, that a threatl h.ts to be cut of 23 threads per inch, 

1 2 OQ 

the pitch being * 1 4 " tnm With a difference of 

1 h 25 ',1 25^ 


34 * The Calculation of Change-Wheels 

0-05 mm pet thread, the diffeience at the end of 10 thicads 
will be equivalent to one-half of the thiead, whilst at the end 
of 23 thieads, the diffeience will amount to the en the thread 

The foiegoing example clearly demonstrates that only 
fractions diffenng by some thousandths of a millimctie, or 
some ten thousandths of an inch, can be employed 

How can such an appioximatc fraction be arrived at? 1 

Regular piactice often enables one to find a fi action which 
approaches veiy closely, without the assistance of any method 

In one of his note- books the wntei found a fraction which 
had been discovered, apait fiom any method, foi the cutting 
of a 3 7 mm. thiead on a lathe with a leaclsciew having a 
pitch of 10 mm 

For this thread theie were no change-wheels, for a wheel 
in which the factot 37 appears, which is indivisible, is not to 
be found among an ordmaiy set of change-wheels 

Foi this reason, according to the notes in question, the 

fraction -~ was chosen, for which change-wheels could DC 

f , 
found, since 

7x11 35 X 55 

-_- ---- -= - - 

208 13 X 16 65 x 80 

q 7 y? 

Seeing that the difference between -~~ and ~ is simplv 

10 208 L J 

the difference between 3-7 and 3*701 = o ooi mm., so that 
aftci 10 thieads the diffeience is still only croi mm., which 
may be considered neai enough for all practical pin poses 

Such gi oping about in the dark, however, is not at all 
methodical, can take a veiy long time, and, finally, may not 
lead to any actual icsult. 

The compound fraction, however, supplies us with a leacly 
means of discovering a fi action which appioximatcs suffi- 
ciently to pei nut the obtaining of what is piactically an 
accurate thiead. 

Suppose the fi action to consist of two numbcis, the 
numciator and denominator of which arc both positive 

for Strcw-tittting on Lat/ies 35 

Let these numbois be lepiesented hy A and B, and A > B 
This can then be tepicsentcd 

^ =s rti 4- 'l /', < B 01 B > ;-! 

Taking the icveisc of the last-named fi action, the i eduction 
ran then be huthei continued, 

K , ''' ^ 

= (h 4- , /j < ?i 01 i \> >'> 

Continuing fuithei 

which can be continued ad infimtum, and can thus be 

/-, " '-.' 
in which 

),,<>,< i <>' ''// i > >w 

The (juotionts /?,, ^ 3 , ^. </*, ire termed indicators, 
By substitution can be obtained 


a i + ', etc., etc. 



' - 1 

1) 2 

36 The Calculation of Changc-Wheeh 

If ._ o, then the numbci of tenus is finite, in which 

case the fiaction is detcimmable, in that it can finally be 

divided without leaving a icmaindci. 

If the pioportion be mdctciminablc, and cannot con- 

sequently be cxpiessecl by a fiaction with exactness, then 
theic will be no end to the divisions, in which case the 
number of teims of the compound fraction will be infinite 

Every indeterminable numbci may be ief.aided as the Until 
of an indefinite, non-iecumni> fiaction The /////// of a 
lepcatmg decimal fiaction is a determinable ptopoition, ei>, 
the limit of 0-3 is J. 

To apply the foicgoing to a definite fiaction. 
(i) Given A > B, for example 

To expiess the fiaction as a compound Itnclinn 

6961 6961 

4- i 

+ i 

3 + 

93 1 

-f i 

A + " 

-h 1 


The indicators aic thus I, 2, 3, /j, 5, 6, 7. 

for Screw-cutting on Lathes. 37 

Consequently ^ 7 as a compound ft action =. 


6+ * 

2+ ---- 

* + 
j \ 


(2) Given A < B, foi example. 

To express -^ as a compound ft action. 

H3 _. _1_ _. i 

355 355 , _i6 

113 3 ~ l " 113 
+ i 


If ..- <; I, the first mdicatoi can then be expiessed by o, 

in which case the mdicatoi s will be 

o, , } and T \ r , 


113 i , ,. 

- = o -f- as compound fraction 

355 * 

(3) Express the compound fraction 4 as an ordinary 

fraction. 3 




A = 4 + X == 4lt 

3 + 2 + \ -3* S 

38 The Calculation of C//ti/i^i'-ll iin A 

(4) Expiess the compound haetion \ as an oidnuiy 
fi action ' 



3 + 

= A + 1 

T ~ (i 

A 1 03 

1? is thus = 
B 215 

The genual fonnula can now be expicsscd by putting 
Icttcisin place of the figures given in the, foic;,Mmi;,> examples. 

Given the compound daction a, determine the ouliiuuy 
fraction b 


given that 

a i 

b = 2 

c = 3 
</ = 4 

a + 



a />( d \ a b \ </ (/ i t 
d but I b I d 

bed -I- /; -f d 
cd -I T 

l Cd H- 1 

</ " ;/ 


lu d s= 34 
= 2 
^/= 4 
c d &= 12 

bed =, 24 
b 2 

- the 1 nuuu'iator. 

30 = the denominator. 

so that in this ease the value of the fi action A ' 


Jor Sit'cw-in'titig OH Lathes 39 

1^)1 any given value of a, l>, t, and d, the fi action can be 
imim-diaU'ly detci mined from the ft action 

a hi, d + a b -f a d 4- c, d + I 
b t, d -f- b + </ 

To take the level se Given the otdinary fraction 

(a b -f i ) 6 4- d 
/>i+ i 

dcU'imnu' the compound d.iction. 

(a b + I ) ( 4- _. rf d <? + t -f a 

/^ + 1 /><+! 

= a -\ = indiCtilois. 


'I'he indicators ate thus a, b, and c 

(liven that in the foiegomg fi action the indicatois have 
the following value a ~ 2, b 3, c = 7. 

Then revcisnig the oidei of things in the foiegomg 

(a b + t) c + a _ (2 X 3 -|- i) 4 + 2 _ 28 4- 2 = 30 
bt -\- i 3x4+1 12-fi 13' 


The inclicutoih foi the h action ^ aie thus 2, 3, and 4, 

The foregoing consequently pioves 

(1) That uvcty detcrtiiinablc fraction may be expressed as 
,i finite compound fraction. 

(2) That every finite compound fi action maybe expiessed 
us a (Interminable fi action. 

Compound factions may be divided into: 
(a) Symmetric. 


4<D The Cah ulation of Change- \l r hceh 

If teims and compound fi action be cxpiesscd as 

B = (^" ai a "~ 



is tcnned a symnictuc compound fi action because the in 
dicatois end in the same oidei of sequence as they began , and 

is teimcd a penodic compound fi action, bee , uuse the uulu atots 
a,,, a it rtj occui peiiodiudly In both cases the ntunbei of 
terms is infinite 

77/# Findmg-ont of ApptMimating 

Whcncvei the factois of a fi actual, act 01 ding to winch a 
thread is leqimed to he cut, aie not rcpicsented by the 
change-wheels belonging to the lathe, it is impossible, us bus 
ah each/ been dcmonstiated above, to cut a theoretically 
acctnate tluead, but an attempt can be made* to discover a 
ft action, the value of which appioaches that of the teal 
ft act ion so closely that the two may be icgardecl as piaetiealty 

Such an approximating ft action can bu found by resolving 
the fraction into a compound fraction, and toinunatlnp, this at 
the second, thud, fouith, fifth, etc., indicator 

Foi example 

A . i 

= ^ 4. 

jor Scrctv-uiftmg on, Lather 
Koi the fust quotient substitute 

then the second quotient will be 

l 'a ~~ , _ T 

= rfi -+ 

CJa tf a 

the thud quotient being 

"'' ss a 4- ' = rfj ( 

?a 4- 4- i 
j rt a 4- i 

, etc. etc. 

die the i educed uppioximatecl fuictions, the 

values of which uie alternately ^icvitei and smallci than the 

A A 

value of , and they appioach moie and mote closely to ,,, 

which may consequently be regarded as then limit 

The ^i eater the ntimbei of indicatois, the smaliei the 
difference between the approximating fi action and the exact 

value of 


The following connection can be established between the 

fractions and the indicators :- 

p __ ji 

a 4- i) H-i 

3 ^a 4- PI 

t on.suquently 

P., =3 ,, P,, 4- PI and Qj = ., O a 4- Qi- 

It follows, thcrefoie, as u geneial rule that 
P, * a, i P n -i 4- P-a and Q = a n Q M ,1 4- 
and this can be applied in the following mannoi : 

4 2 The Calculation oj 

(i) Given the fi action ^ Deteimme the compound 
fraction, i.e. the indicators, and find an approximative fraction 

A _ 51 , , i 


B 16 ' ^ I a, 5 

3 - 3 

B Qi I ' * 

A P a s X 5 4- I 1 6 , , i i , 11 i 

._ = ,..- = 7 J = . limit appro.ichc'cl still tlosci 

B g a 5 5 

A P, 3(16) -1-3 ^i 
- = ^ = , = ,,, the exact valiu' 

B Oy 15 + 1 16' 

Given the fraction 


A = 3370 __ i 

B 399 2 + I 




p t 

,,= , l '". 17 

U j 2 

{?,, = 4 < P a = , s (^^ 4- i) + <t, ~ 76 ( li - j + i ~- 9 

17 y,-^= - ( , 2 

i V 

= 7 Pj = a 4 (,,( 1 tf a + 1) +!^ + 1) = 549 Q t , t (,.^ f J) \ a 

a D = 6 ^ P r , = a R (4 ( fl 'i (^iss-t- + ii; t a + 0) + <*n (i<^ + 

6 x 549 -I- 76 

Ur, = n (4 (03 #a + i) + ^a) 4- rt^ + r 

6 x 65 -i- 9 -- 399 

The appioximating fractions are thus 

3 j; 76 549 3370 
i' 2 ' "9" ' 65 ' 399 ' 

Jor Screw tutting oit I,athc\ 43 

(;) Deteunine the compound h action ami the appioxi- 
matinj; h.utmiis <>( the mmibei 2 "j 182818 28459 

A 718 \Si,S2.s i $<) i 

= ^ j 4 - - 

I + 

p, 2 
(J, ~ I 

- t P., ~- 2 X 1 + I - } Uu -* ' 

~> \> < v l > y / >sxiii- > "=r 

1 ., : X { t >' - i VJ,i X I | I > ( ) i 

' Pi I I 

--- i P t - i x s | ^ n <Ji ~ i x H ' - ! ( > ~ I 

P., '<) 
= i P 6 ~ 1x114-8^-19 (](,=. i x .1 -M ~ 7 ( ) f) ~ 7 

-()' ^' = K ' 7 

(4J IMctminu the approximating ft actions for the niunbci 


7 , 

7 + 

,,-=? 15 P-.-333 (J.,= H)6 

rt 4 t I'^SSS LU-^3 

rt & 2tjJ P B 103993 (J B 33102 

</ sa i 

^ t =s j etc. etc, 

<I H = 6 

44 The Calculation o/ C/iange~ll'hccls 

The approximating; fi actions aie, consequently, 

3 22 333 ^55 

i i x- j i Cl t I Hj. 

i 7 i 06 113 33102 
Fiom which the following can be detetmmed ~~ 

Axiom i The diffcience between two sm cessivc apptoxi- 
matmg fi actions is, the signs not beiinj; taken into ronsideta- 
lion, equal to the unit divided by the pioduct of its 
or, in gcneial, 

v = r "- 1> " + I = ( ~ J) " 
* Q* ( J + , <J,(j + ," 

Should theic also be tlnee successive ap 
h actions, 

1> i) |> 

1 n - i * n * u \ t 

<}-/ Q/ <> + / 

the fiist will then be gieatei than the second, the seunul 
being smallei than the thud, etc 

Example (sec page 42) . 

A__ 3^70, 
B - 399 ' 

the approximating fi actions are 

8 17 70 5/19 
i' 2' 9' 65 

v _ -^ H-i - 1 
; '~ 2 ' 18' 585 

Axiom 2. The diffcience between the exact value of the 

fi action B and one of the approximating fi actions will in- 

vanably be less than the unit divided by the pioduet of thts 
denommatois of this appioximaling A action and those follow- 
ing, and also less than the unit divided by the square of the 

/or S(rcw~tnfting on Lathe* /|5 

denommaloi o( the fi act ion undei consideiation , 01, in 

A _ P, ( - i r 
( - i )* ( - i )" 

<.>,.<], <V 

A __ \\ i 

u (] < (} ((_-!- o,, ,) 

Application A __ 3370 

H ^u() 

I 1 , H P, _ 17 I', ___ 76 P 4 

( ji = i ( j- ~ -' ( 2i u ( ]> 

3V ~ l '7 <s ' 

3')y "^ - ! ~ 599 < ~" 2 

30U " ' < t (H 

137 () _ J 7 < ! 

< etc. etc 

2 ^ 16 

From which it follows in order to obtain an appioxi 
mating fraction, tliflenng only a millionth pail fiom the exact 
vaku 1 , UK* (k'nomiiuttor must consist of at least /) figures. 

The tliffcrcncen between two successive api>roxiniating 
ft actions become ronliuually smaller, and are alternately 
positive and negative. The difference appionches //, and 
nmsequeotly the limit of the approximating fr.ution to the 

r A 

exact value of .,. 

46 The Calculation o/ (. h<ui(~\\'hcci\ 

By mtcipolation anothei fraction can still be found behvcen 
two appi oxi mating h actions 
Gencial teim 

P a P 4- V 

1 '*>/ " H - T l * >t - 

By taking in place of a, the values i, ', 3 


_ t ), othci fi action s c<in he mleipolated between ' "" and 

, both of which foi m an mcicasnu; 01 diminishing thaw, 
as they both have the same sign 

(i) Required, the interpolated fractions between ' = l ^ 

i ^4 549 ,- ,1 f , A 3370 . 
and , . = > J of the fraction ,, = ' ( naie A M 
O, 65 U 399 Vl ^ ' ; 

an = 7 ^,~, = 6, 5, 4, 3, ? and i, 

\\ <f tt l\- t + P_, 
Q* ^iQ-i 4-o 

l' = 549 O* =- f)5 

P =17 ( ) = > 

\\ = C> x 76 + 17 _ 473 i j . _ 3 X ;(. I- 17 \|5 
() 6x9 + 2 50 () >t ~ 3 x 9 + i ;>9 

P __ 5 x 76 + 17 _ 397 P,, _ n x 70 + I? ,( Mj 

y 5X9 + 2 ^J7 O ff ""* : x () + > *" ;jo 

r w = 4 X 7^ 4- 17 _ 321 

Q 4x942 "" 38 (j,, i x 9 + > 1 1 

The factions 93 *<*) ^45 3*1 W d7J 

11* 20' 29' 38' 47* 50' u< lims " t- 

twccn the fi.ictions 7 and y^, which nre anpioxiniathi" 

2 (J " * 

fractions of .- . 

jor Strew-tutfuig on Lathe*. 47 

P H 

(2) Requned, the interpolated fractions between ' = 

and . ' = of the same fraction 

O ! 9 

IV = <i,, 1V-1 + IV ~, IV ^3 x 17 f <^59 

<j tf/^-.-f <J- U "3x24-1" 7 

IV =- 76 <J = 9 }} ^ 2 x 17 -4- 8 = 42 

() ? x 2 + i 5 
IV- ,= i7 (],-,- 2 

IV = i x 17-4- <s = 25 

lV- 2 == U,-. = i <> 1x24-1 3 

5 o 42 3 5 
consequently, the appioximutm^ fiactions -;_, " " , he 

between ' and 


/ 3 >*J 

i 9 

I /( ) 

(3) Rcquuccl, the interpolated Iraetions ' = and 

l j r, _ 3370 
UB 399 ' 

ft f) = 6 #,,_,; = 5, 4, 3, 2 and i 

IV = 3370 U* 399 

IV -i = 519 Q-i = f) 5 

IV _ 5 x 549 + 7C _ ^2i IV 2 x 549 H- 7<^ _ "74 

()/-"" ex <>K 4- o 3^4 ( ) 2 x 65 -j- 9 "~ 139 

*^' J * J ' S <J*' r *" J ' * " 

IV 4 x 549 + 7^ _ 2272 IV _ i x 5/19 4- 7^> _ r >-5 

<J* ~ 4 x 65 + 9 "" 269 (J^ ~ i x 65 -H 9 " 74 

IV _ 3 x 549 + 7^ = 1723 
(j w , 3 x 05 4- 9 204 

, . P . 625 1174 J723 2272 . 2*S21 

the approximating fractions , , ' , ' ,uui 
thus he between / and 

48 The Calculation of Change- Wheels 

Application Determine the compound fraction and the 
approximating fiactions of the numbci 2*539954, so as to 

5*4 12 ' 7 
er piopoiio ~ 

r L 

inch in cm 

2*4 12 ' 7 

obtain another piopoition as 01 ~ foi expressing the 

r L 100 50 1 tS 

A __ 2539954 = ^ 
B ~ io 6 

3 + 



1+ ' 

14- ' 

2+ ! 
I I 

The mdicatoi s aie consequently 

2, i, i, 5. ! > 3> $, 2, 3, 2, i, i, etc. 
1235 28 33 127 1049 ?22 S 
i i i 2 ii 13 50 413 876 

The following and the approximating fiactions can he 

obtained by interpolation between fractions 2l and ' 2 ^ - 

n 50 



Qo = 50 

^ - 


Pr, = 


QB= 13 

"o - \ == 

2 and i 


1 ^ 


(J<i I I 


__a n 

* W I "T" J W 2 

2 x 33 4- 28 

=r 9<l 



Qx-, + 0,,-.,,"" 

2 X 13+11 

~ 37 

i x 33 4- 28 


i x 13 4- n 


By interpolation between s = ^3 (UK j P? _ 1049 

Us 13 <Jv 413' 

following can be obtained 

for Sirew-tuttmg on Lathes, 49 

IV _ a,, I 1 ,,, + \\, . = i X 127 + 3} = 160 
<l a,, (} , 4-Q w . , i X 50+ 13 63 

_ 2 x 127 + 33 __ 287 

2 X 50+ 13 ~ H3 

_, 3 x 12 7 + 33 = 4M 

3 x 5 11 4- 13 163 

_,} x 127 + 33 ^ 54i 

4 x 50+ 13 213 

__ S X 127 + 33 = ^8 

5 x 5 + 1 3 -"63 

__(> X 127 + 33 = 795 
0x50+13 313 

__ 7 X 127 -)- 33__ 022 
7 x 50 + 13 ~~ 363 

so that the following approximating fi actions can be found 

\\ , ID jo 160 287 414 541 668 795 

between ^ and ^ v,, fjj , n ^ ,63' 2,3-63' 3^3 

and ^. 

t's in Condition. 

(i) It is icqunotl to cut 3- 1 , thieads pet 2}J, in Leacl- 
icw ^ huh pitch. 
Fitch to he cut ~' () Lcachcicw J inch pitch 


*' x 2 - ^^ ~ ' 13 
x If - } ""x4"" 28 

No wheel with ,13 teeth is to he found, and the numbci 
f |3 ih indivisible. It will thus be neccssmy to find an approxi- 

mating fraction, 

'H I 

fi action = ( ^ -- i 4- " j 

1 + 

5O The Calculation of Change-Wheel^ 

Indicators . i, i, i, 7, 2. 

a, = 

ff a = I P (J = I X 1 + I = 2 ' 

F, ^ 

tf a = I > P a = i X 2 + I = 3 O , = I X 1 + l = '2 ( )', ~ 2 
ff/i = 7 P 4 = 7 x 3 -f 2 = 23 O,, = 7x2+1 = 15 o* - *,", 

a rt = 2 P 6 = 2 X 23 -I- 3 = 49 Or, = 2 X i >; + 2 = 3 ' ( , 6 ~ , 
^ %,,,() .' 

Intel polating between ^ and 

1* rt w P_, -f P w n 3x23-1-3 26 , 

-" = " J = =- is obtained 

() ^O"" 1 + O"~* 1 X 15 + 2 17 


^ =i 5312 which is o 0045 l css tnun ^ lc Actual d.ietion 

J *** 

26 ,. 

= i '5 2 94 0-0063 

This cliffctcncc occuis in eveiy 2 thtcads, so that the 
actual diffcicncc pei pitch is <inly o 00225 

appiofichcs most closely to these two, so that the 
wheels will consequently be 

49 _. 7 X 7 _ 70 X 70^ 
32 4x8 40 x Ho* 

(2) Requhed to cut a pitch of 3'7 mm, 1 ,cru Iscrcw 

10 mm, 

Solution : " . 

Theie being no wheel with 37 teeth, and the number 37 
being indivisible, an approximating' fraction will have to he 

for Screw-cutting on Lathes 

Compound fiactum 



2 4- 

1 f 

2 4- 

i -f 

Imhcdtois ,ii( v thus o, 2, i, j, 2, i, 3. 

-(!-,- 'Q,., 

-- ? , P. O X 2 4- I 1 Cj a = 2 

! ., i 

.- i i P, =- J X i 4- o = i ' O ; , = 1x24-1=3 

Pi sr ^ y r 4- i j= ' 

x 4 "-" * f'x i ^r i ~"" . 

<7< ~ 2 

X 3 4- i =7 

i x 7 4- 3 ~ 10 

P 7 -= 3 x 10 4-7 = 37 

O,, = 2 X 3 + 2 = 8 

(J f) =2x84-3 = 19 
y (! = i x 19 -f 8 = 27 
Q 7 = 3 x 27 -I- 19 = 100 



Qi l 

"O j 

Q = 2 
p t = I 

Q.i 3 

1^ J 5 

4 ^^ O 

Q * ~ s 

T"^ *7 

P fi ___ 10 

Qe ~~ 2 7 


The appioxiraating fmction ~ 3'74> which only chffcis 

* J / 

from the actual ft action by o*oo/{ mm. per tluead, may thus be 
accepted ior <ill practical pui poses, 

IO _ 2 X 5 _ 20 X 50 

27 ~ 3 X 9 ~ 45 X 60" 

(j } The Proof of the Smn. 

The comparison thai 6*5 in, 165 mm., or an adopted 
fraction, is not perfectly accurate. Should it be desired to 
find out to what extent the fraction which has been arrived 
at, and, consequently, the thread to be cut, deviate, this can 

E 2 

52 The Calculation of 

be done, when a mctiic thiead to be cut on a l.ilho having 
an English Icaclscicw, by multiplying the numeiatoi ul the 
fi action by the pitch of the leadsciew in mm The 1 pio 
duct thus obtained should coincide with the' piodiut ol the 
denommatoi of the fraction anil the pilch to In cut, t.e 
numcratoi X pitch of leadsciew in mm. = denominatoi x pitch 
of thiead to be cut 

Numeiatoi, denotmnatoi and leadsciew pitch lx nu known, 
the pitch of the tin cad to be cut ran consequently be delei- 

On pay;e 28 the h action ^^ has In en detei mined !oi 
a pitch to be cut of 2 mm, and a leadseiew ol 2 tlneads 
pci inch 

The piodutt of numeiatoi and loadsuew pile h m mm. is 
thus 26 x 12-69975 01 26x 12 7 = 330*2. This piodmt when 
divided by the denominate of the d action \\ill ;i\i the pitch 
in mm to be cut with the* wheels deleunnu-d in, thus, 
330*2 : 165 =2 ooi mm The pit< h is consequently cxutt 
to within O'OOI mm. 

7X J3 _ 91 
11X20 ^20 

is given on page 29 foi a pitch of io\ mm , with a lead screw 
of I in. pitch 

91 x 25-4 2311 'A 

J J ^ J ' 


22O 320 

The pitch is theiefoic exact to within iroofjj mm. Both 
these differences may piactically be re^atded as of no tonse- 

In the case of a lathe having a metric lujulsciew on which 
ICnglLsh. thiead is to be cut, the denominator should be 
multiplied by 2 -54 The numeiator when divided by the 
piocluct thus obtained, gives the pitch to be cut in inches 

On page 24, the fraction for cutting f threads pei inch 

with a leaclsciew of 10 mm. pitch is mven as l(> ^ , 


If the denominator be multiplied by 3*54, the* result will 

be 1<5s - = l6s 
6x65x2-5/1 990' 6" 

for Screw-tutting on Lathes. 53 

K.uh pitch cut is thus o- 1065656 in 

The exact pitch = \ in. = cri6 in, so that the thiead 
cut ihlfeis only by o-oootoio in 

Note, that when cutting motitc thiead with an English 
Icadsciew, the thiead cut is a fi action too coai se, whilst, on 
the contiaiy, when cutting English thiead with a metnc lead- 
sciew, the thread obtained is a fi action too hue. 

(/&) Fixing vp the Wheels 

It is not always possible to fix up the 4 wheels in the older 
of sequence given in the examples. 

FIG 10. FIG, ii 

The following h action may, for example, occui . 


in which the wheels must be placed as per Fig. io 
although the wheels 30 and 55 cannot mesh. 

The fraction can, however, be ai ranged in another order of 

54 The Calculation of Change-Wheels 

sequence, viz > , which makes fixmer up possible (see 
55 X 125 t. i i v 

Fig. H), but care must be taken that the wheels of the 
numerator are nevci placed in the denommatoi, 01 vice vci w?. 
Should simple changing about of the factors in numeiator 
and denommatoi, 01 one of them, be impossible, the fi action 
is then resolved into the lowest possible factors, and .mother 
combination of wheels sought for, which will give the same 
pioportion between numciatoi and denommatoi, as, for 

30x50 _ 2x2x3x5x5x5 _ 30x40 3OX.]o 
55x125" 5x5x5x5x11 ""55x100 50x110 

(/) Thread-t,uttiiig wit/i Double Compound Tram. 

Should it be necessaiy to cut a tin cad consideiably coaiser 

01 finer than that of the leadsciew, it can easily happen that 
the necessaiy wheels aie lacking. 

For example, to cut 56 tin cads pei inch, leadsciew 

2 threads pei inch. 

The fraction is * = IO x 1 5 A wheel with 10 teeth is 
56 70x120 

lacking 1 . If the numeialor and denominatoi ol the fi action 
aic once again multiplied by 2, a wheel with i/jo teeth is 
obtained m the denommatoi, which us also not at hand. 

In such a case, the numcratoi and denommatoi of the 
fi action aie icsolved into 3 factors, as, foi instance 

2 = 20 = 2X2X S _, 20x25x30 
56 560 5x8x14" 70x75x80' 

Example: To cut 48 threads per Inch. Lcacksuew 2 
thieads per inch. 

Solution. 2 = 2 2X 2X 5 20x25x30 
48 480 5x8x12 60x75x80 

/or Screw-cutting on Lathes 55 

(m) The Cutting of Left-hand Threads . 

So fai, it has been implicitly taken for giantcd that only 
n^ht-hand tin cads had to be cut, it can, howevei, happen, 
though not often, that a left-hand thicad has to be cut. Foi 
this pmpose, the leadsciew must lotate in an opposite diicction 
to the lathe-spindle This is obtained by connecting up an 
idle wheel at will, In double ttansmission, a fifth wheel (idle), 
chosen at will, may also be mtioduccd 

A number of lathes have been constiucted of late which 
rend ei the connocting-up of an intermediate wheel un- 
necessary. With these lathes, all that is icquned is to shift 
the leveise-plate at the hcaclstock which reverses the move- 
ment of the pinions which duvc the change-wheels, thus 
causing these wheels and the leadsciew to lotate in an opposite 
direction This is a decided impiovcment, as theie is not 
much space to spare when five 01 six wheels are on the shear. 
With a double compound tiain generally the larger numbei 
are only small wheels, but with foui wheels, however, every 
proportion is possible, so that the placing of a fifth wheel can 
sometimes be vciy troublesome. 

56 The Calt His twn c/ (Xv //<;? 

CliAITlCK 111 


TniSKK aic diffcient foiw. ol thread, a h-w of uhhlt aie 
illustiated in Fi^s. 12 15 

Fig. 12 shows the Ver thiead in its "rwtal fuuu, which is 
conM.1 noted in different types, ,uul is inns! often met \\ith 
Ki^. 13 illusti.itts the stjuau' <u ll.u (Ijit'.u!, the st-aion of 
which is eithoi u S(|u,uc 01 a ii<;ht an;;lc, and wliit It is much 
in use for luis^'i dismictois and >ai',n thuads. In IM*| 14, 
the tiapexium tlucad is smi, Ihc Mrtin nt whu'h is a tiapc 


I''J', 12, 1,3, 14, is. 

aium, much in vogue {or the hvuKcrcws of lallu-s, the \\uun 
being also a tiapeyfum thieati, FiV,, K; ,s the immd thu-ad, 
ioimed by the iiiiej, section of snniYuvU's, 

Voiy little need be said with ivft-nnut- t*t flu* last thm- 
types, fur which it is inipossiMc to ,spak of any om-'in, 
the form of thu section beitis; d pnnirnt tm u'n uinstanrrN 
and detei mined by each individual at will 

Different vaiiutieH, however, exist uf Ihe Vei* thrt-.ul. 

(/;) Tj>/>?\ f 

The type chiefly employed ts nnlainly the Whitworlh 
system ; Mg, 16 shows llus constmction, 

The depth of the Whit\uwlh threail t.s uqual tu u'<i,j of 

for Strew-tzitmi> on Lather, 


tin pitch, ihc sides of the thtead foi mini; an an^le of 55" 
with top and bottom lounded tlnough \ of the line //, 
drawn pctpemlicnlai fioin the- apex of the tnanide to its 
base, tin* ladius of toundini', being equivalent to O'l (3 h 

Not only is ihc sectional foiui oi the Wlutwoith tlnead 
definitely fixed, but also the numbei of thieads pei inch foi 

FK. 16 

h 0,96 S 

all dumeteis up to and including 6 inches, and this has been 
fixed at from 20 JJ threads per inch. 

The sectional loim i.s piecisely similai foi the finest as well 
us the coarsest threads, and it is for this icason that the exact 
dimensions and strength of the tlnead aie determined by the 
simple determination of the outside diametei. 

I >l.ltlUll 1 ,ll 

Hi) (.torn 




<> as i 

1 7^ 



7-.!^ -A! 




c>*<52 , '&) 


1 6 


it'll 54 




i."7o ' '39 

9 <JX 



'5*^7 "S 1 



l i 

ID-US , ' 




^,"^2 '73 




i<5'4 ' 'H4 

21 '33 



28 'W , '04 

3 7 



it '75 i U 7 



DitUtu'tet at Duimetei of 
liuLtuin 1'hiead 


mm in. mm 


34'(J2 l'i(> 2fJ'4<*> 


38 'I I'^ij 32 '6S 



i -37 35-28 


4 ?J 

j'4y V7 N 



i 71 *nM3 

3 i 

57' 15 

rtjj 49-02 

55 '37 

f0 4S 

No of 

per inch 



70 ' 

3-03 6() Ho 3| 

58 T/ie Cahulatitw <>/ Chatty II 7/rt-A 

Table 1. mves the v.inous dimensions of tilt' Whitwoitl 
tin cad 

A Whitwoilh thiead of in tain dimensions can also hi- ui 
on a lonsuletabiy lai^ei otitsuU 1 diameter, the evut stu i n!' ( tl 
of the tin cad beim> fixed by simph del CM mining; whirl 
dimension of Hie VVIutuorth system is leqimed 

T.ible I t'.ives not only the outside- diameter, but also tlu 
cliametci at bottom oi llnead, so that the In ij'Jit nl the thiead 
can be at lived at by subtia* tin, 1 * the lattei itom the iotmei, 
and dividing the diffeienee by two 

When cutting tlueads on the latin, whu h deviate in 
ihumetei from this system, it is necv.saiy to know the depth 
of the thiead both ioi tutting msule and outside thieads 

The depth of the tlnead < an also be airived at by a simple 

Foi this piu pose, just look at Ft'i^. in. By diawim; a 
peipendiculai horn the apex of the Uian;le t a ni'hl-antjled 
tuangle is formed, the smallest am;le of \\huh is equal to 
55"-r2 = 27" 30'. 

Tuny. 27" 30' =O'^L Thciefbte, if the lunj| side of the 
ught-any le = 1, then the short side ^ 0*5.^, and the base tif 
the triangle of 55" ~ I '04 

This base is, however, equal to h, i.e. the pitth. 

Whence it follows that h : S = i : i '<v|, or o'*)ti : L 

The ical depth of the tlnead is, however, only ;| li, So 
that the latio between the depth oi the Urn-ad aiul the 
pitch is equal to j h : S =r (o-yO x f t ) : i cs orj t | ; i Jj // tlm.s 
equals O'6/i S, 

If we take the outside tliametei 1), the diaitietei at the 
bottom of the thiead 4 and the pitch S then, d ^ 1> ** 
2 x 0-64 S, or d I) i '28 S 

The gas thread universally adopted by the pipe tuuk*, given 
in Table II., Is also according to the Whilwtnth sysluin, and in 
1903 was also adopted ua the .standard thread for pIptjH and 
fittings for gas, water, and steam by the AsstMUtitlon of 
German Engineers, the Assocititioti of (icnnan Fltimber^ the 
Association of the German Central Heating Industry, and 
the Union of German Pipe Manufacturers, 

for Scmv-tutluig on Lather 


On the othct hand, in the autumn of 1898, an attempt was 
made by ,i niuulu'i of mllnential associations oi Continental 
en<>iiH'eis, assembled in confess at Zuiich, and including, 
amongst nthets, the Association of Gennan Kn^meois, the 

1 \III I II \VlIll\\URII! SlUlWlNl. Tllkl',Al>. 

1' vti_tn il 

in i nitu ! HI <mn in mm in nun in 

h } 17! ^j u 71 ^( 8'^ 28 lA }8 ) l 882! 47 81 I 7<>S 44' 

J j o 35 -qi8i} i->j hi u 41 "9 ill 1i'-7 ' u *' i S 1 i3 r 904 48 

li ' 9 S' '<)S(t't(> f>7' '$8iji4 9i 19 14 44 <|S- f H7 S 2 J 9} 49 


jj '15 H7J 9<>"-'*9'i 8 1 20'' 
,J 'it, os'f oj. lad 4l' '<).19 '( 

14 2 50 8 U vi 7 59 f>i- 33 5f> 6 ^ u 

1 1 2} s?* 1 ^- 3 5'^7 f'S 7- 2 "l7 i ()2 7^ n 
H 33 03 5 '> 7^'-'^ S8a 73 =7 

?, 2.-".v t uS<HO J 1-1)9727-87 14 .',' (ly-K 
i 2tj'J t ^iKjhiV4 t ' 192 i. 'H ii 3 762 

J47 247i M 

4*5 ^8 SM 3 (lS 
912 99 V'3 795 

|3 75;* (J 5 

-rjaj i 74$ 

41-91 1-^338-95 H 4 101 (i 4339100-2 4223 

4 1/32 i -028 41 '36 II 

79 5 1 u 

8S 5i II 

9 f ' 39 i 1 

[07" 26 II 

AIII.K III. ~S. 1 Ti 



l)i imttu 

at lUiHnill 

1 Main 


1 ll.Ulll'U I 

ui 1 hicail 

))i mi 


nl Tluuwl 















if)' 75 



41 ' 5 



> 7 



18 -75 



45 '5 


i *a^ 







^ ^S 


i j<; 

7 37 



-53 'J 












56 c)i> 





3 S 






i 7S 






f>" 5 

63 '55 






33 ' 








*l S 










39 'S 


The Calculation of Change-Wheels 

Swiss -Association of Machine-Tool Makcis, the Society 
the Encomagemcnt of National Industries, etc., to replace 
Whitwoith system, which is based on the English system 
measurements, by a metnc thicad, and it was unanimously 
decided to adopt the S I thread (" Systeme International ") ' 
as per Table III 

Owing to the universal application of the Whitworth 
thread, the innovation makes but little headway, thong" I" 1 
especially of late years, this system is being more and 
used on the Continent, especially by the Automobile 
for thieads cut on the lathe. 

FIG 17 
The construction and form of the S I thread is mvera in. 


Figs 17 and 18. 

The apex is an angle of 6o n . The section is consequently 
an equilateial triangle 

Hence it follows that the peipendicuhu h, dropped from 
the apex to the base, is equivalent to 

The truncation equals | /t t so that the thiead has a 
of 0-75 A, 01 o 6495 S. 

for Screw-cutting on Lathes 


Whilst the Whilwoith tin cad beais not only at the sides 
but also at the bottom, the S I thiead, on the contiary, has 
a play at the bottom of, at the most, ,',, //, equivalent to the 
half tiuncation, the rounding of the thiead is equal to the 

A/vv : \/v:.;n 


, D D, 

\ A Y ' 
^/ ,* t 

FIG 18 

play, the radius of the loundmg in this case being j l () // The 
rounding and play amount, as is generally accepted, to at 
least ^ /{, Loewe stakes ,in aveiage foi this, and fixes the* 
play and loundmg at ^,, h 

The outside diametei of the male-si lew is thus smaller 
than the diameter at bottom of the thread in the nut, and 

FKJ, 19, 

vice versA, the diametei at bottom of the thread of the male- 
screw is smallei than the outside diameter of tin cad in the nut. 

62 The Calculation of Change- Whech 

If we take the play a, then the actual depth of the tin cad 
of both male-sciew and nut equals o 75 // + u If we fix the 
play at its maximum, equals T \, k, then the height equals 
0-0625 /; + 0-75/2 = 0-8125 /*, 01 o 703625 S = ~0'7S. 

The Lowenheiz thread (Table IV) is in j>onei,il use up to 




at Bottom 
uf Thread 



it llolttiin 
uf I In cad 

Diiin 1'itih 

Ui unitu 
it Bottom 

Of IllK.ul 







nuu mill 



o 25 

o 625 

2 6 





S'S y 



I 2 

o 25 

o 825 






6 i 



I 4 

o 3 

o 95 

3 5 





7 i i 



I 7 

o 35 

i 175 






8 J 2 




o 4 


4 5 





9 i * 



2 3 

o 4 

i 7 






10 i 4 



a diametei of IO mm foi instiuments of eveiy description, 
especially in Gei many and Swit/eiland, and in scicw works, 
the sciews are almost exclusively made by this system 

The construction of the Lowenhci/ thie.ul is shown in 
Fig 19. The apex is 53 8' 

This angle icsults from h ~ S. The tin cad is titmcated 
flat on the outside diameter and at bottom with a J truncation, 
so that the ical depth of the thread is = 0*75 //. 

The Sellers thiead (Table V.) is an Ameiiean thread, con- 
stiucted as pei Fig 20 

for Strew-cutting on Lathe*, 


Nunilii r >{ 

I'll! < U.I' 

)ll 1 UK ll 

I'liu ul 
pen mi h 

1,1 Mini ter 

Ntuuhii ol 
Hiu id, 
pu inch 

















The apex is t in an^le of 60, so thai the peipcndiculai f { 
(hopped fu)in the apex to the base, is again = o*866S, The 
tlueacl is Hat-faced at bottom and on the top with J trunca- 
tion, consc(|tiently 

t = ,"] fj and 0'7S X 0'866 = O"6495 S. 

The thread which resembles the S. I. tin cad veiy much 
has, however, no play and is divided accoiding to English 

Although largely displaced by the Sellers thread, theshaip 
V thiCiid still exists and is used in America, (See Table VI.) 
The section is an equilateral triangle not truncated. 

6 4 

The Calculation of Change-Whccl\ 

The B A. S (Bntish Association Standaul), as per '1 able 
VII, is an English tluead much used in Kn^land foi sne\\sof 
small diamctci, especially foi electiu fittings. Tin 1 - a|H \ t 
Fig 21, is an angle of 47 \ n The thiead is tninratt'd, and 


Number of Number ot Niiuiln i of j Niimlii i nj 

Dnnit-tcr rineads Dmiitlc.1 I liu uU Diumui Iliiivid hiuiiilii J'iin ul , 

ptr mi h 

pi.r UK h 



mi h 





n 1 








1 fi 
l ii 



I 7 i\ 




* H 





2 ! 








I ft' 



3 1 







1 5 



top and bottom aie rounded, leaving the depth of the 
equal to o 6 S 

In addition to the foi cooing, the Delisle, Sauvnje, Acmt 1 , 
and Thuiy systems aie to be met with. 

The total number of thiead systems exceeds fifty, hui only 
the seven most used have been treated of ht'ic, 




l r m etei> 6 5'347 4'i 3'6 4 va 2 H 2'S '. i-Q 1-7 r 3 i 
^ i 0-90 810 730-660 Spo'S^o^Ko'o'io-.- 

Jot" Xi 

on Lathes, 

((} Si i ew-c lifting Tools. 

A tool used ioi sucw-cutlino m iist fust and foiemost be 
peifet tly tine. It is not to he looked upon as an oidtnaiy 
tool, noi may it be mound on a stone which does- not mn line 

When tuttim; deep threads, whethei they be V 01 squaie, 
it is always advisable to use sepaiute tools loi toughing and 

The cutting an{le must be about 70, whilst the tool must 
not !r pointed 01 somi-t nculai, but flattened at the cclqc 
(\ f li\\ n 2 and ?3), as othenvise the angle will not be true, and, 

.it the same tune, it will be impossible to nrind the tool 
accurately The tool must not only stand on its edge in the 
alible B, IMJ,' 23, but the sides A A must also have cleaiance, 
The atisde in which the thread lies on the work has also to be 
lakvn into etmsidetation, and the line AB, Fig, 24, must tun 


Fie.. 2<j, Fir., 26. 

at the same angle. Suppose that a I in. pitch has tu be cut 
on n diameter of 2 in. Then, imagine C D in Fi^, 26, to be the 
angle a I which the thread lies on the woik, the line AJBof 



The Calculation of Change II '/r<-A 

the tool, Fig. 25, must thus run patallel to the line (' 1) in 
Fig 26 This is still moic evident in the i ase of sqnaie 
thi^cads with a coa ise pitch, Fig ?7 In this i ase, tlu <kuan<e 

on the sides of the tool must be diOcient, The diametei of 
the tin cad on the top, as ,ilso the angle of the tluead thete 
is indicated in Fig. 28, that at bottom of the tluead in 
.^ Fig 29, a and b being the ehrumferenee,, 

t and c the pitch, which is tlu* same for both, 
and thcic aie consequently two angles* 
The hypotenuses if/ and <* show the un^lc of 
the tin cad til top and bottom. If the 
cleaianceol the tool is tont'ct on the 
will be mcoueet when at the bottom. The 
hteepei the pitch, the moie noticeable this 
will be. The tool must have more elearanr e 
on the right-baud side for bottom than ut 
the top, but less on the left-hand side, 
The tool must consequently be ground in 
such a manner that the tight hand side will 
have enough elearauu' at bottom of the 
thread, whilst the eleanmre foi the loft- 
hand side must conr.iu with the angle ul 
the top, that is to say, foi a light-hand thread, as in Fig. i*7 ; for 
left-hand tin cads 01 for internal threads the opposite conditions 
will exist in icgard to angles. The tool must menrdingly be. 

It C-i 

FIGS. 28 AND 29, 

pound as indicated in Pu,> 27, A R being- the slope of the 
iKjht-hand side ot the tool, A C on the left-hand side. The 
iippi-i cutting smfart' of the tool must um squat e on the line 
A l>. When <uttim> an inside linht-lund thiead, ey;ciythinj> 
is i excised, is li^ht-hamled becoming left. 

Foi a Vee tht cad, the tool must be <round in accoulance 
\\ith the aU",lc- of the system ol the Ihioatl, ft need scniccly 
be said that this must not be left only to eye 01 the lough 
estimate of the opeiatot, A tauge should be piovided, as 


pel 1'V, 30, ;ivnv', the pierist* un^le. And yet, notwithstand- 
ing that it is lai nou k diHieult foi a woikman to judge an 
tun;/! 1 \uth llu e)t- thun to ^uess i ceitain /etigt/i, and no one 
would eun think of peimittin^ an opeiator to estimate a 
u'lt.un length without usinu his rule, it is- an exception when 
the opnatot K ptovided with n suitable an'le gauge. 

// /\ nttt'i'ty iw/xnA't/'fc that a thread can be trite when llie 
tfuttttH JIM juried //!<' rW/i' J the ttwl with his naked eye. 

This {juuuc 
purpose 'kvrn 


a .second, and not less important, 
the tool be gnound to the precise 

It is htill possible to out a wrofl^ thread, for the tool 
must be M pkeed in the holder that an imaginary line diawn 
pinpcndteularly tnn the apex of the triangle to the imaginary 
ImstMUUht alho fall perpendicularly on the side of the cylinder 
on which the thread is to be cut, Not having this gaugc, 

I? 2 

68 The Calculation of Change-Wheels 

the opeiator judges with his eye the position m 
thinks the tool should be placed But the most 
woikman can make a mistake, it is not possible ^f 
to be ti nc if the tool has been placed with the 
in the position which might icasonably be 
collect, and this is afteiwaicls checked with an 

FIG 32 

it will almost invaiiably be found that the position r 
The icason is that the two lines forming the ang>-lc . 
vciy shoit in piopoition to the othei lines of the t:o< 
being consequently deceived 

In Fig. 31, at A, Is shown the manner of gauj^inj 
to which a lathe centic should be tinned, at B, tl 

FIG. 33 

which a screw thi cad- cutting tool should be 
C, the concctness of the angle of a scicw thieacl -<\ 
In Fig 32, the shaft with a sciew thiead is ^vip 
held between the ccntics of a lathe. By apply In 
as shown at D or E, the tin cad tool can be set zxt: 

for Sircw-cnttwg on Lather, 69 

!<> the sh.ilt, and then fastened in place by the boltb m the 
tool post, tlu-U'by avoiding nnpeifect 01 leaning thieacls 

' n I'" 1 ", 33* <-t V and G, the mannei of setting the tool for 
< lilting internal tin cads is illustrated 

(d] Cutting t/ic Thread 

As pievumsly stated, it is always advisable to begin 
cutting a IhuMcl has anything like a deep curve with a 
loughing tool which us at a cutting point and which need not 
be {Ljiound piecisely to the angle. 

The thiead should afteiwaitls be gone ovei with a 
finishing- tool When engaged in cutting shallow threads, the 
tool can cut on both sides at the same time, and it can be 
put exactly on the duection of the shaft. With deeper 
tin cads, i.e. quick pitches, this is no longei possible. Cutting 
with both bides of the tool at the same time causes it to snap, 
the thiead is tough, and veiy often it is impossible to continue 
working; the tool should, theiefoie, woik but one side at a 
time, should fiequently be set slightly m a paiallel duection 
to that of the shaft, and cliiectly theie is any play between 
the tool and the thread, it must again be set squat e on the 
direction of the shaft. Each time that the tool has gone 
completely ovei the thread, it should be withdrawn and again 
set in the oiignul position at the commencement, though 
increased with the amount cut at one passage 

For this piupose a gi actuated collar is provided to the 
feed screw by means of which the traverse movement can be 
read, and by which the tool can be set in the exact position 

every time. 

"The opciator foi racily got out of the difficulty by maiking 
the position of the screw spindle with a piece of chalk, 

On lathes of up-to-date consti notion, the giaduated collar 
is now always to be found on the sciew spindle. 

A very practical consti uction is shown in Fig. 34- 
Advantage is hue taken ol the movement of the two half- 
nuts when opening and closing, to withdiaw the cutting tool 
irom the cuive, and mis vcrsd, back again to the exact 

The CaknlafKni of C/ian^e-ll 

position, so that instead of having to cany out vaiious opeia 
tions at the end of the thtead, a simple movement of a 
handle is till that is icquiied. 

The constiuction is as follows Ovn tin* (\\o half mils 
which move unclci the carnage in the .same du tit ion as tin 
cioss-slide, and aie opened and. closed by a double n;;hl- aiul 
left-hand sciew, is placed a Pi-shaped slide lived mi knobs of 
the upper poition of the half-nuts, 'I he su< w spindle ol the 
cioss-slidc fits in the uppci poition of this slide on the one 
side by a turned up ed^e, and on the othei by huk-nuts 
The scicw spindle must consequently follow the movement o! 
the slide Holes aie dulled ii ( idU thtoii",h the ptojetthu 1 , 


v r , ,, * ' I I "\ 

j- -^,J,ir""r J^' ",, * j f ^ r U ^) 

paits of the half-nuts, and the slide, A steel pin lit-, t fnsrly 
into these holes. Oblong holes, in which the pin has play, air 
boied in the carnage for same. 

Before beginning to cut internal or evtenul llm-suK, tlu* 
pm is set in the foiemost 01 hindmost nut, so that the halt nut 
thiough which the pin is placed Ls coupled with the slide In 
which the scicw spindle fits, and umseqiiemly they imtsl 
follow the movement of the half-nut in que.stion lont-ther 
with the cross-slide and tool. It is wotked as Jollmvs: As 
soon as the tool has anived at the end ol tin- threat!, the 
half-nuts of the Icacl-scicw are opened and by this means the 
tool is withdiawn fiom the thread The caniagu j then 

liftin on 


n tinned l>\ hand hy means of the pinum, the tool set so 
imuh faithet in with the sue\v spindle as it is desiiod to cut 
drepej, and tin- half-nuts aie closed again This causes the 
tool to lesnnie Us oiigmal position, only cutting the uiatcnal 
st much deeper as it has been set Luther in by hand. If no 

thtead is lo 1> ( nit, the connection between the slide and hall- 
nut is Inoken by withdrawing the locking-pin, and the slide is 
coupled to the uoss-shde by insetting the pin in the hole 
boied tlmniyh both slides 

When screw cutting, this anangeinent ies.ults in a decided 


saving of tune, l>esiilcs preventing the possibility of mistakes 
arising from inserting the tool eithei too far or not far 
enough In, 

There should be an outlet for the tool at the end of the 

72 The Calculation o/ C/MH^C l\ f Jn.cL\ 

thiead If the diametei is sufficiently lai;>e to pet mil of it, 
an entiie circuhu gioove should be tinned, Ki<> ^5 If, ioj 
some reason 01 othci, a citcul.u gtoove is not possible, a 
suitable outlet, as pel Figs 36 39, must be dulled foi vee ot 
squaie thieads Bcfoic commencing cutlin<j ( , the tool should 
be so fixed that it will antvc just at these holes. 

It was foimeily the custom to tetuin. the uuna^e when 
the tool had gone over the tlne<id, by tevoishi", the movu- 
rhent of the lathe But with the ptesenl-day eonsiuu tion 
of the lathe, by which it is possible to letnin the carnage 
quickly by hand by means of the handle, the h, Ul-nuts ate 
opened and the caniage leturned by hand. If the thiead bcttu; 
cut is of the same pitch as, 01 an aliquot pait of the pilch of 
the Icadsciew, the half-nuts can be chopped into engagement 
at any point of the Icadsciew without any difficulty, the tool 
always ictuimng to its pi ease position in the thiead This 
is, however, not so when the nmnbei of thiead s pet inch aie 
uneven 01 bioken, and othei means must be adopted to ensnie 
the tool ictuintng to its precise position in the gioove. Con- 
sequently, when slatting to cut the Ihtc.ul, a slop, ot mm king 
line, is placed on the bed, the half-nut closed and a ehalk line 
diawn on top of the leadsctew, and anothu i hulk line nt the 
fiont side of the chuck-plate. When the tool bus none oyet 
the thiead and the can tug c has been teUnned by hand as Jar 
as the stop or the line, the head spindle is turned lound till 
both chalk lines are again in then original position, the nuts 
closed, and the tool is once moie m its pieuse place in the 
path which has, just been cut, 

This compaiatively troublesome and punnUve manner of 
working is done away with, if the c;iniaf;e is piovklcd with a 
thiead indicator as shown in I ( "i^ 40, 

The following is the principle of this uttat hment ; A stwvll 
worm-wheel runs on the leadBeie\\ and by moans ol a pinion 
gcunng, causes an indicator to move on a (iicular indox- 
plate. All that in now nuees-saiy is in iuli* the position of 
the indicator at the starting point, allot which, tlu: half. nuts 
can be closed, mid the tool will come pieeisely in the path 
each time the indicator resumes its original position, 

for Screw-cutting on Lather 73 

(?) The Cutting of Double or Multiple Threaded Screws 

The cutting of double 01 multiple threaded sciews causes 
<i ^<>od deal of tioublc, as, in addition to cxeicisiny ouhnaiy 
uue that the thicud cut is tine, anothct most impoitant point 
has to be taken into consideration, viz that the setting of the 
tool is also exactly equidistant The manner of working is 
smiilai to that foi a single thiead, but caie should be taken as 
fat as possible that when cutting a double thiead the spindle 
wheel is divisible by two, and foi a ticble thiead by three 

After the fust incision has been made to the lequned 
depth, the tool must be shifted exactly to the centie between 

Fltr 40 

two threads for a double, and to one-thud of the intei mediate 
space for, a treble thiead The distance the tool is to be 
shifted should, however, nevei be mcasuied off, as this can 
novel be exact, but must be obtained by mechanical means, 
either by turning the woik-piece while the Icadscicw is 
slationaiy, or by turning the leadsciew while the work-piece 
remains stationaiy, If a double thiead has to be cut, one 
of the teeth of the spindle wheel coming between two teeth is 
maiked with chalk, as also the two teeth which the tooth in 
question engages., After this the spindle wheel is bisected and 
this tooth is also chalked ; the spindle wheel is then icleased 

74 The Calculation of Chaugc-Wltceh 

from the wheel it engages, the spindle is given half a turn by 
hand, so that the opposite tooth comes between the two 
maiked teeth, and the two wheels rue once moie engaged 
If the spindle wheel is not divisible by two, then this must be 
found on the wheel on the leadsciew, but the pitch of the 
thiead to be cut must in this case be taken into consideiation 
Foi example A double thieaded sciew of 4 thieads pel 
3 inches is to be cut on a lathe with a leadsciew of 2 thicddfc. 
per inch. 

The fi action is -i-= 3 = 7 *> . 
\ ^ 5 

The spindle wheel is, howevei, not divisible- by 2, and at. 
the factor 3, which is indivisible by 2, will invanably be found 
in that wheel, 4 wheels aie used so that the factot 3 can be 
placed in the intei mediate wheel 

75 _ 100 X 60 
50 ~ 50 x Bo 

If there is any reason, for instance, with heavy lathes not to 
turn the spindle but to shift the carnage by turning the lead- 
screw, this is accomplished as follows foi the above example 

Pitch = | in The carnage must thus be shifted ,"/ -f- 2 = 
| in , the leadsciew has a pitch of ^ in., and so must make 
3 1 = 1 revolution, the wheel of the leadsciew has So 
teeth, and consequently 80X7? 60 teeth must be moved 

If the same pitch is to be cut on this lathe but foi a thtee- 

thread, then the first-mentioned wheels, ?5 arc the best to 


use , the wheel with 75 teeth can be divided into thiee, and 
25 teeth turned each time. 

If it is desired to move the caniuge, this must be moved 
T-T- 3 = i i"-, the leadsciew make 4 i evolution, and the wheel 
with 50 teeth be moved 50 X J => 25 teeth. 

For example, To cut a pitch of ij in. Double thieaded 
screw Leadscrew I in. pitch. 

Solution IL = J 5 = I0 x 6u 

I 8 40 X 80 " 

jor Sit'ciu-cut/iug on Lathes. 75 

Poi ,i double Ihieaded sciew, the spindle wheel is divisible 
by \ 

\\ -i- -3 = J~; in Thi.' lead sci ew must thus make { (j ~- l 
} [\ revolution, 

|[| x Hi) = 75, The wheel on the leadseicw must thus be 
moved 75 teeth, 

K \*amphv~ To cut 6 thieatls per 15 in, thiee-thieadcd 
suew, Leadsuew | inch pitch. 

c i , If' 15X2 75 X 80 

Solution ' JUi_=: > = ' J 

\ () 30 x <\o 

Poi a thiee-thie.ul, the spindle wheel can be divided into 
3 X 25 teeth 

The Cciniagc must be shifted \, 5 3 = il m ' so thal thc 
leuclscicw must make ,] -f- J = V' i evolutions 

The wheel with 30 teeth is placed on the leadscicw, and 
30 x \l } - 50 teeth ate moved = 50 -r 30 = J i evolution and 
20 locth. 

(/) 77/6 1 Cutting of vet y Coarse Thread. 

When cutting coarse tin cad, a difficulty may possibly 
occur which will lequire cuicful consideration When the 
tin cad to be cut is considerably codiser than that of the lead- 
set ew, the movement of the leadsciew must be appreciably 
quickened, There is, however, a limit to this, and that is the 
H-sistante olfeied by the teeth of the goat -wheels If the 
pitch is too coarse, these will bieak off The extent to which 
the pitch may be mutased depends, rmtmally, entnely on 
the stien^th of the wheels supplied with the lathe, Generally 
speaking the pitch may .safely be a four-fold of the leadserew, 
anything exceeding this being attended with considerable 
danger and the off-chance of the teeth breaking. 

hi order to permit thread to be cut which is many times 
toaiser than that of the leadscrew, a gearing can be attached 
to the fast headstoek, as illustiated in Fig. 2. 

The wheel 15 can be set in connection with the small geai- 
whuel of the double back gearing. If then the lathe mns with 

7 6 The Calculation "/ Change- 1 ! 7/< , A 

double back gea,, the ..ilu, "I -P-l l-euveen UH ,..,- Pulley 
and lathe w,ll lx- I K, 'll IK-IWC-,,. .lu- .-1 . , ... ul 
the cone-pulley pm.on 2 ,, ,u,,l the u'h,, 1 ,u w,l U,,,,,,,!, U 
4 .evolutions l<> i of Ilu- sp,,,.IU- Inllu-.-.s,.,,!., ll,,,...,l ,s foiu times auisci th..n the l,.,Kuew, Iheu ,, .1 ,.it,.. 

of I ; i between the t-han^t; \vlnvis, whil ,( tui ,i ihir.ui i-s^JH 
times as coaise, Lheie is only a ratio oi i; TaUm", ,f uii\i* the teeth die stump, etioii-'Ji UH a talus ut I ; ,|, aiul that 
the leadsctew has i in pitiii, then u \ x.j X | K HI, pit'h 
may easily be cut in this l 

lor Si moulting on Lathe\. 


One of the in \\i-st designs fot sc rew~cuttm^ is that of the 
litMhl, v-Muth.n Msicm, \\hieh, ), y HUMUS O f attain of ocais 
p!u<vd umlei ,md ,U the side oi the hcrulslock, icndeis it 
l" IN '' !blr to (Ul 'i nuinbei o| thio.icls o( thfToicnl pitches 

)s ~v*;,'v.v {'"' ^'^^(tWMJSSff MS f /////$/>$, 

, i ' ' l V '\NN\\V. '11 * v-if^jMc "V 1 i *W 

H 'TfT ... ! ' ^ ' ' i! i! ^ !-i 


FfC 42 

with* nit the iHTt'ssity of fixing different ch<mge~wheels. 
("halloo wheels as they have up till now been imdei stood 
JH t'nnnv{}tii with the htthe, luive been entirely supuiHcdcd 
On u Ltlht 1 piuvided with the llendey-Norton system, it is no 
lon;;<M ju'tt'ssary to fix up or lake off change-wheels, the 
various whooK being .simply and solely geared tip in the space 

78 The Calculation of Change /77/6rA 

formed between the spindle and the leadsciew by the shifting 
of handles The calculation of change-wheels is consequently 
a thing of the past 

But, in this woik which tieats of the whole question of 
screw-cutting in an nbndi>ed foim, a descitption of this 
system, which will ceitamly come moie and inoie to the fmnt 
in the stiu<lcfoi economical tools, and has ahead} been vciy 
largely adopted, must not be missing 

" 4 " , VL *> m^'T^s 
.- /A X^m^i^ 

\ ', ;,'?:' . \ ^ ** 

Aruuigemenl o{ wheels m a L<xl|;i> anil Shipli y hulu, the dutt 

On a lathe of this description, scicw-t uttin^ has Ix-cn 
i educed to its simplest possible foim, A drver \vtu*Kaitn 
may, it is tine, be quite capable of e,duil,itin } ", the \vlu k cis 
leqmred to cut a certain tin cud quirkly, and cun possibly 
reckon it out m his head, but even so, llu- utlual fixing up 
of the wheels seriously interferes with the strady pu>^rcs of 

for Screw-cutting on Lathes. 79 

the woik, whilst the difficulty is at once doubled whenevci 
turning, chilling, and thiead-cutting have to be pcifoimed 
periodically, as, with so many lathes, the attendant chcum- 
stanccs ,ue such that it cannot be aridiigccl foi all at the 
same time 

The lathes undci discussion ate constituted in such a 
manner that a gieat vancty of threads can be cut without 
icquuing the fixing up 01 taking off of a single wheel 

In the eailici constiuctions of this type of lathe, theic was 
mvanably one gieat drawback, viz. that the numbei of 
pitches which could be cut was comparatively small (10-12 
pitches), but this numbei has now been extended to fiom 
40 to 44 diffeient pitches. 

The foicgomg illustiation (Fig. 43) shows the complete 
anangemcnt of the wheels 

This gives a cleai view of the bed, the fast Deadstock 
having been icmovcd foi the piuposc 

The anangement of the wheels consists of two sepaiate 
groups of wheels The fust gioup (9-11 wheels) is placed 
undei the headstock, the second being in a closed box 
attached to one side of the lathe 

The action peifoimed by a wotkman in geanng up the 
wheels foi the cutting of diffeient pitches is cxtiemely simple, 
so that aftci a biicf explanation it is sufficiently cleai even to 
a novice, tmd it can be executed so quickly that not more 
than fiom 10-20 seconds aie icquiicd to change the wheels foi 
anothci pitch than that foi which they weie geared up. 

An index plate is affixed to the gear-box, which is given 
on page Bo In its exact si/e. 

A handle with pointer is placed undci the plate. This 
pointei can be moved over the cntue length of the index plate 
and set in the middle of cithci of the foui divisions of the 
plate. This handle is connected with the wheel indicated in 
Fig. 41, by the number 862, winch accoidingly moves this 
wheel with it , whilst nuclei the holes in the headstock the 
numbers I 8 or i~ii appeal, accoidmg to the dimensions of 
the lathe. 

If, lor example, it is leqmred to cut $2 thieads pei inch 


The Calculation oj Changc-irhcch 





I lid', 

'I lid 
















1 6 



I 1 
J, 1 


s 1 





1 i 






I J 

80 to 40 

40t() 20 

Pi A I I? 

It) to 

the pointci is placed by means of the handle in the middle of 
that division in which the numbei in question appeals mulct 
the letteis Thds. (Tlneatls), in this paitieulai ease, in lh" 
second division on the light hand '-ide. On the same line on 
which the number 5] appears, the fignte f t 1 ? to be found. 
The handle on the head stock is now plated m the hole above 
the figure 6, and the wheels me than <>eaied up foi euttitif* 
the desired thicad For all other thteads appearing on the 
index plate, the ptocedme is identical The topmost handle 
957 ls pl acc d m the highest 01 lowest position, uivoidhi^ as it 
is desired to cut left or light hand thread 

We will now pioceed to give a detailed description of the 
constmction of this gcanng. 

Wheel 968 (sec Fig. /|2) is fast on the lathe, spindle and 
engages wheel 922 (Fig, 41) whunevet right -hand thread is to 
becut In this case wheel 933 is idle. Kot , i left-hand threat!, 
wheel 968 engages 923, and wheel <).L' i.s caused to total e by 
wheel 923, so that the direction of movement is just the 
rcveise to that in the fust case. Both wheels run loose on 
studs fastened in plate 920, and arc- shifted by the middle 

for Screw-tutting on Lathe*,. 81 

handle Wheel 922 engages wheel 955 which Is fixed on 
shalt 992, which is consequently biought into motion, This 
same shalt ^52 nnpaits motion to wheel 959, which, by means 
of a key way, can be moved in a tiansvcise diicction by the 
handle undei the fast headstock Wheel 959 engages 961, 
which can be geaied up, by means of the handle ahcady 
icieued to, with all the diiletent wheels 651-659 mulei the 
fast headstoek, which wheels aie all fixed on shaft 662 , wheel 
961 consequently impailhig motion to the shaft. Wheels 666 
and 067 ate also kej ed to shaft 662. Wheel 862 (Fig 41) 
movable by a keyw.iy, is mounted on the leadscrew. Con- 
sequently the motion of blufl 662, to 'which the gem-wheels 
ate keyed, is ttansmitted to wheel 862 by one of the wheels 
606 01 (167, vid two sets of double wheels 905 and 906, both of 
which setsaie identical 

This tiani of geais can be seen in the detailed drawing* 
Ktj- /jl, to the lelt ol the side view of the fast headstock. Jt 
should be noted that wheels 905 and 906 aie coupled, but that 
each set is independent of the othei, and can consequently 
lutate at diffeient speeds , this is, moieovei, appaient with the 
whole turn of geais, seeing that, whilst wheels 666 and 667 
also coupled, and each engages one of the sets 905 and 906, 
the latter obtain vaiious speeds. This tiam of geais gives 
four diffeient speeds between shaft 662 and the leadsaew 

Wheel 606 engages 905 and 906 on the right Wheel 667 
engages 005 and 906 on the left, 

By moving wheel 862 on the leadsciew (this wheel is 
also to be seen in the illustration, Fig. 43), and by changing 
handle ydj, which tuins on shaft 662 and to which at the 
same time the two sots of wheels 905 and 906 are keyed, 
wheel Kfo can be placed in lorn different positions, i, 2, 3, 
and 4, (See detailed dt awing, Fig. 41.) 

Wheel {>(; =- 906 and wheel 666 = 905 =s 862. The 
proportion of 007 to 906 .= r I, of 66(>to 906 =2 i, so that 
If wheel %j engages 905 on the light, the speed of shaft 662 

* ^ V* ^ 

i,s doubled, seeing that ** 2, 
** i X2 

If wheel B63 engages 906 tm the right, the motion ol the 


The GittM/tithw of C/Miigt- //7v'A 

shaft is tiansmitted without nn)' vanation, am I ulu-t i <,<< n 
the light simply serves as an idle wheel, li ,H* > t n-; ,t",e<, 'in;, 
to the left, tlieie is a double i eduction in sped, U ,Sn* < n^a-M 
906 on the left, the diminution is lour tunes is 'iieat. <*tm 
scquently, if the handle on the fast headsttu k is set t <j nut" 
No 9: 

With the pmnti'i in uihmw J, ^ iiUh<', pt-t n It 
*'. '/' 
). *4 

4, 2K 

will tn nt 

In this manner, with n wheels on shaft fh\ 4 j tltllei 
pitches can be cut. 

The swing plate of the headstoek IN hutlun * tun 
structed that, by netting up oius wheel, the spt'ttdu! tlt< lt%w!- 
scicw can once more be doubled, or by removing fin* ?..wu 
wheel, it can be reduced to half as slow ugahi, so that all flu 
threads appearing on the index table can now be ujl, with 
double or half the number per inch, The reseive hle tn the 
swing plate can be cleaily .seen in 1%, 41, ulM* to i,>,*j, 

In the foic^oing illnstraticniH, Fig, 44 glvc;H the ebljit- 
lion foi fine threads, Kig, 45 for coarse threads whilst W|t, *|6 
shows the position of the wheel 955, 

for St rew~i ntting on Lathes. 


Tin- iisu.U ;e.utm Is: Wheel 968 engages 922, and 922 
ur;.tn's u^, the wheel on shaft 9152. For fine thieads, 968 
\ U'si'tes OJ~, and c> >2 engages o?3, consequently 923 engages 
t> {, n'luch IN ,i double whet I with 9?$, the proportion between 
them being i 2, Finally, 923 engages 955 Wheel 955 
d*H% nut engage <)2\ hut is moved a wheel's width to one side 
(See Fig, 4] 6) 

MfH^iAft TffltCtOS Of /NOKK 

Frit. 46. 

Km rtmisi^ threads, qf>8 engages 922, and 92 
i.tiu't|iwntiy ^1's 923 and 923 engages 9^5. Fiom 
,i t',m*lut ionsith-ration tf these two comb!natkm& foi fine and 
MMi.r tlsie.nN, it will he seen that wheels 924 and 925 on 
tiu* OIH- suit;, atul wheel 923 on the other side, are mutually 
iftfti*h.ini',*Hi foi the two cases. 

Sfai It !WH ontyhocn multiplexor fractions of an inch, 
in both, which nmia !>o cut in thin manner. Should it, how- 
rvu- ht Ht'WN-ary to deviate boredom for any special pitch, 
tlwi thrr,tits than ihoscof the KnjUHli system can also be 
,-irt by 4 u-ruin proportion between the two wheels 924 and 

0-5. 3992